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**Constant, Linear and Non-Linear Constant, Linear and Non-Linear**

Types of Functions Constant, Linear and Non-Linear Types of Functions In this section we extend the concept of function to include functions with special properties. The notion of slope, introduced as average rate of change, is introduced and illustrated. Types of Functions 7/9/2013 1

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**Constant, Linear and Non-Linear**

Types of Functions Constant Function Each member of a set (domain) maps to the same member of another set (range) A = domain B = range a k b Constant Function We begin our coverage of function types with constant functions. These are functions with a singleton set as the range. As the illustration shows, every member of the domain is mapped to a single element in the range – the constant functional value. In the illustration, the function F relates (maps) every element of the domain A to the single element k in the range (set B). So, the functional value never changes, i.e. it is constant. For constant functions the domain may be of any size, even infinite. The range, however, is a singleton set containing just the constant value of the function. This relationship is illustrated in the next slide. Note that F(x) = k for every x in A. In other words, F = { (x, k) | x A } c { } F = ( , ) a k , ( , ) b k , ( , ) c k , … Types of Functions 7/9/2013 2 Types of Functions 7/9/2013 2

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**Constant, Linear and Non-Linear**

Types of Functions Constant, Linear and Non-Linear Constant Function Examples f = { (-4, 3), (-1, 3), (2, 3), (4, 3), (7, 3) } x y x y 2 3 4 3 7 3 Constant Function Examples In this example we examine several aspects of the function f, defined over a finite domain, { -4, -1, 2, 4, 7 } . As the illustration shows, we can view f as a set of ordered pairs, or since the set is finite, we can list the pairs in a table. The pairs of course are exactly the ordered pairs in the set view of f. We can then use the ordered pairs, either from the set or from the table, to plot corresponding points in the plane. This set of points is the graph of f , in this case a discrete graph. Note that the graph is not the function, but merely a representation of the function, as is the table. The reason for this is that not all functions have graphs and not all functions can be listed in a table. The set of ordered pairs, however, can always be used to represent a function and we therefore take it to be our definition of what a function is. These facts become clearer when we examine the second example which describes the constant function represented by the symbolic form f(x) = 2 for all real x. We note that, while there is a graph (a “continuous” one), there is no table representation, since the number of ordered pairs is infinite. Thus, the domain of the function is an infinite set, while the range is the singleton set { 2 }. Domain = { -4, -1, 2, 4, 7 } Range = { 3 } Types of Functions 7/9/2013 3 Types of Functions 7/9/2013

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**Constant, Linear and Non-Linear**

Types of Functions Constant, Linear and Non-Linear Constant Function Examples f = { (x, 2) x R } x y y = 2 Domain = R Constant Function Examples In this example we examine several aspects of the function f, defined over a finite domain, { -4, -1, 2, 4, 7 } . As the illustration shows, we can view f as a set of ordered pairs, or since the set is finite, we can list the pairs in a table. The pairs of course are exactly the ordered pairs in the set view of f. We can then use the ordered pairs, either from the set or from the table, to plot corresponding points in the plane. This set of points is the graph of f , in this case a discrete graph. Note that the graph is not the function, but merely a representation of the function, as is the table. The reason for this is that not all functions have graphs and not all functions can be listed in a table. The set of ordered pairs, however, can always be used to represent a function and we therefore take it to be our definition of what a function is. These facts become clearer when we examine the second example which describes the constant function represented by the symbolic form f(x) = 2 for all real x. We note that, while there is a graph (a “continuous” one), there is no table representation, since the number of ordered pairs is infinite. Thus, the domain of the function is an infinite set, while the range is the singleton set { 2 }. Range = { 2 } Note: y = f (x) = 2 No table Why ? Types of Functions 7/9/2013 4 Types of Functions 7/9/2013 4

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**Constant, Linear and Non-Linear**

Types of Functions Constant, Linear and Non-Linear Linear Functions Always of form for constants a and b f(x) = ax + b Linear Functions The form ax + b is called the slope-intercept form of a linear function. The reason is that the constant a is the slope of the graph of the function and b is the vertical coordinate of the vertical intercept of the graph, which is the point (0, b). We can take this symbolic form as the definition of linear function. To see what happens when x is incremented by 1, we compute the difference f(x + 1) – f(x) = ( a(x + 1) + b ) – (ax + b) = ax + a + b – ax – b = a Note that a can be either positive for an increase in f(x), or negative for a decrease in f(x). The rate of change of the function is represented by the slope number a, which can be read directly from the symbolic expression if represented in slope-intercept form. We emphasize that the character of the rate of change of the functional values determines whether or not the function is linear. The function is linear if and only if the rate of change is constant over the entire domain of the function. If a = 0, then the slope, and hence rate of change, is zero. In this case the function represents a constant function. Note that zero slope does not mean there is no slope. What if a = 0 ? Question: What properties does f(x) have ? Types of Functions 7/9/2013 5 Types of Functions 7/9/2013 5

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**Constant, Linear and Non-Linear**

Types of Functions Constant, Linear and Non-Linear Linear Functions Changes in f(x) = ax + b Always proportional to changes in x f(x) changes by a per unit change in x Rate of change of f(x) Rate is a, the slope of the graph f is linear iffi rate of change is constant Linear Functions The form ax + b is called the slope-intercept form of a linear function. The reason is that the constant a is the slope of the graph of the function and b is the vertical coordinate of the vertical intercept of the graph, which is the point (0, b). We can take this symbolic form as the definition of linear function. To see what happens when x is incremented by 1, we compute the difference f(x + 1) – f(x) = ( a(x + 1) + b ) – (ax + b) = ax + a + b – ax – b = a Note that a can be either positive for an increase in f(x), or negative for a decrease in f(x). The rate of change of the function is represented by the slope number a, which can be read directly from the symbolic expression if represented in slope-intercept form. We emphasize that the character of the rate of change of the functional values determines whether or not the function is linear. The function is linear if and only if the rate of change is constant over the entire domain of the function. If a = 0, then the slope, and hence rate of change, is zero. In this case the function represents a constant function. Note that zero slope does not mean there is no slope. Types of Functions 7/9/2013 6 Types of Functions 7/9/2013 6

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**Constant, Linear and Non-Linear**

Types of Functions Constant, Linear and Non-Linear Linear Functions Changes in f(x) = ax + b Always proportional to changes in x f(x) changes by a per unit change in x Linear Functions The form ax + b is called the slope-intercept form of a linear function. The reason is that the constant a is the slope of the graph of the function and b is the vertical coordinate of the vertical intercept of the graph, which is the point (0, b). We can take this symbolic form as the definition of linear function. To see what happens when x is incremented by 1, we compute the difference f(x + 1) – f(x) = ( a(x + 1) + b ) – (ax + b) = ax + a + b – ax – b = a Note that a can be either positive for an increase in f(x), or negative for a decrease in f(x). The rate of change of the function is represented by the slope number a, which can be read directly from the symbolic expression if represented in slope-intercept form. We emphasize that the character of the rate of change of the functional values determines whether or not the function is linear. The function is linear if and only if the rate of change is constant over the entire domain of the function. If a = 0, then the slope, and hence rate of change, is zero. In this case the function represents a constant function. Note that zero slope does not mean there is no slope. Types of Functions 7/9/2013 7 Types of Functions 7/9/2013 7

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**Constant, Linear and Non-Linear**

Types of Functions Constant, Linear and Non-Linear Linear Functions Rate of change of f(x) = ax + b Rate is a, the slope of the graph f is linear if and only if rate of change is constant Linear Functions The form ax + b is called the slope-intercept form of a linear function. The reason is that the constant a is the slope of the graph of the function and b is the vertical coordinate of the vertical intercept of the graph, which is the point (0, b). We can take this symbolic form as the definition of linear function. To see what happens when x is incremented by 1, we compute the difference f(x + 1) – f(x) = ( a(x + 1) + b ) – (ax + b) = ax + a + b – ax – b = a Note that a can be either positive for an increase in f(x), or negative for a decrease in f(x). The rate of change of the function is represented by the slope number a, which can be read directly from the symbolic expression if represented in slope-intercept form. We emphasize that the character of the rate of change of the functional values determines whether or not the function is linear. The function is linear if and only if the rate of change is constant over the entire domain of the function. If a = 0, then the slope, and hence rate of change, is zero. In this case the function represents a constant function. Note that zero slope does not mean there is no slope. Types of Functions 7/9/2013 8 Types of Functions 7/9/2013 8

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**Constant, Linear and Non-Linear**

Types of Functions Constant, Linear and Non-Linear Linear Function Examples f (x) = 2x + 1 , for x A where A = { -2, -1, 1, 3, 4 } x y -2 -1 x y = f(x) 1 3 4 -3 -1 3 7 9 Linear Function Examples: Example 1 The illustration shows an example of the symbolic form of a linear function, defined over a finite domain A, as shown. This means, not only that we can graph the function easily, but also that we can list the function in a table. We show via the arrow connecters how each domain element is matched by the symbolic function form with a corresponding range element to form an ordered pair that is then plotted in the plane as a graph point. Because it is finite, the domain can be listed in a set. Similarly, since the domain can never be smaller than the range, we can also list the finite range. The graph in this case is a discrete graph and, while all its points lie on the line represented by y = 2x + 1, the graph is not the line but the finite set of points extracted from the table. Discrete Graph Question: Is rate of change constant ? Types of Functions 7/9/2013 9 Types of Functions 7/9/2013 9

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**Constant, Linear and Non-Linear**

Types of Functions Constant, Linear and Non-Linear Linear Function Examples f (x) = 2x + 1 , for all x R x y = f(x) Domain = R y = 2x + 1 Range = R Linear Function Examples: Example 2 Here we have the same symbolic representation of the function, but with an infinite domain – the set of all real numbers, R. Considering the locus of points in the plane that have coordinates satisfying the equation y = 2x + 1, we see that the graph of the function is in fact the line represented by the equation. So the graph is a “continuous” graph. Because the graph can be projected onto the entire horizontal axis, the domain of the function can be seen to be all of R. Similarly, the points of the graph project onto all of the points on the vertical axis, making the range of the function all of R. The rate of change of the function f is just the slope of the graph, which is 2. So, yes, the rate of change is constant, verifying that f is a linear function. Continuous Graph Question: Is rate of change constant ? Types of Functions 7/9/2013 10 Types of Functions 7/9/2013 10

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**Rate of Change and Slope**

Constant, Linear and Non-Linear Rate of Change Example f(x) = 2x + 1 for x R How does f(x) change as x changes ? Rate of change of f(x) relative to x is x y = f(x) y2 ∆y y = 2x + 1 y1 Rate of Change of Linear Functions We continue our examples with the same function, y = f(x) = 2x + 1, defined on all of R. We want to select a random change in x, by selecting two random values of x, namely x1 and x2. Reading left to right the change in x from x1 to x2 is the difference x2 – x1, which we call ∆x. The change in y, from y1 to y2 we call ∆y, and since y = f(x), we say that ∆y = ∆f . The rate of change of f with respect to x is just the ratio of ∆f to ∆y, that is Rate of change of f We now simply compute ∆y and ∆x and the ratio of the two. The illustration shows the details of the algebraic computation (each student should make sure that he/she understands the details of this computation). The result is that, regardless of the values of x1 and x2 , the rate of change is 2. It should be pointed out that this 2 shows up in the original symbolic form of the function as the coefficient of the variable x. This is not an accident but a property of the function f. ∆f ∆x ∆y ∆x = x1 x2 ∆x Types of Functions 7/9/2013 11 Types of Functions 7/9/2013 11

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**Rate of Change and Slope**

Constant, Linear and Non-Linear Rate of Change Example f(x) = 2x + 1 for x R x y = f(x) ∆f ∆x ∆y ∆x = y2 – y1 x2 – x1 = y2 ∆y y = 2x + 1 (2x2 + 1) – (2x1 + 1) x2 – x1 = y1 Rate of Change of Linear Functions We continue our examples with the same function, y = f(x) = 2x + 1, defined on all of R. We want to select a random change in x, by selecting two random values of x, namely x1 and x2. Reading left to right the change in x from x1 to x2 is the difference x2 – x1, which we call ∆x. The change in y, from y1 to y2 we call ∆y, and since y = f(x), we say that ∆y = ∆f . The rate of change of f with respect to x is just the ratio of ∆f to ∆y, that is Rate of change of f We now simply compute ∆y and ∆x and the ratio of the two. The illustration shows the details of the algebraic computation (each student should make sure that he/she understands the details of this computation). The result is that, regardless of the values of x1 and x2 , the rate of change is 2. It should be pointed out that this 2 shows up in the original symbolic form of the function as the coefficient of the variable x. This is not an accident but a property of the function f. 2x2 – 2x1 + 1 – 1 x2 – x1 = x1 x2 = 2 f(x) = 2x + 1 ∆x Types of Functions 7/9/2013 12 Types of Functions 7/9/2013 12

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**Rate of Change and Slope**

Constant, Linear and Non-Linear Rate of Change We now have ∆f = ∆y = 2 ∆x Change in f(x) is twice the change in x … f(x) = 2x + 1 for x R x y = f(x) y = 2x + 1 x1 x2 y1 y2 ∆x ∆y ∆f ∆x ∆y ∆x = = 2 Linear Function (continued) Having established the rate of change of f(x) as 2, we can now write ∆f as twice ∆x, so that each unit increment in x produces an increment in f(x) of 2 units. Graphically, this means than if we move from any point (x, y) on the graph one unit to the right, we must move 2 units “up” to stay on the graph. This ratio we call the slope of the graph and assign the character m to represent the slope of the line. Thus the slope of the graph is defined to be just the rate of change of f(x) with respect to x. Since we will be using the concept of intercepts, we define them here. The graph of f will generally intersect one or both axes. The intersection points are called the intercepts for the graph, and hence for the function. The intercepts are often associated with an interpretation of data represented by the function and its graph. The horizontal intercept (or x-intercept in this case) is the point (x, 0) that is the intersection point of the graph and the horizontal axis (x-axis in this case). Note that the y-coordinate is always 0. The vertical intercept (or y-intercept in this case) is the point (0, y) where the graph intersects the vertical axis (or y-axis in this case). Note that the x-coordinate here is always 0. For a linear function, whose graph is a line, we can glean information about the slope and both intercepts from the equation of the line, as we shall see in the next slide. … at any x Types of Functions 7/9/2013 13 Types of Functions 7/9/2013 13

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**Linear Functions in General**

Constant, Linear and Non-Linear Intercepts from Formulas Suppose we have a line L with non-zero slope m and slope-intercept formula y = mx + b x y L Vertical Intercept Slope Intercepts from Formulas For each point (x1, y1) on the line, the coordinates must satisfy the equation of the line. That is, we must have y1 = mx1 + b and this is true in particular for the intercepts. In slope-intercept form the general linear equation is y = mx + b where m is the slope of the graph and b is the y-coordinate of the vertical intercept ( 0, b ). The horizontal intercept can also be read almost immediately from the equation as ( – b/m, 0 ). Each of these intercepts can be found by remembering that the intercepts are points on the axes. So, solving the equation for y when x = 0, and for x when y = 0, produces the nonzero coordinates of the vertical and horizontal intercepts, respectively. Note that the vertical intercept lies on the vertical axis, and the horizontal intercept lies on the horizontal axis – a point often missed by some students. Also note that intercepts are points not numbers. (0, b) Horizontal Intercept ( ) b m – , ( ) b m – , Types of Functions 7/9/2013 14 Types of Functions 7/9/2013 14

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**Linear Function Example**

Constant, Linear and Non-Linear x y Find intercepts and graph for y = x + 2 Note: y = mx + b m = 1 and b = 2 y-intercept : (0,b) = (0,2) (0, 2) Intercepts and Graphs from Formulas In slope-intercept form the general linear equation is y = mx + b where m is the slope of the graph and b is the y-coordinate of the vertical intercept ( 0, b ). The horizontal intercept can also be read almost immediately from the equation as ( – b/m, 0 ). In this particular case, m = 1 and b = 2. So we know that the graph is a line with slope 1 and that it intersects the y-axis at (0, 2). We could use this information to find another point on the graph, in order to draw the graph. Or, we can find the horizontal intercept as shown and use that point, along with the vertical intercept to draw the graph. The intercepts are convenient because they are easy to locate, always having one coordinate that is 0. Once two distinct points on the graph are established, we simply draw the line through the two points. Note again that the vertical intercept lies on the vertical axis, and the horizontal intercept lies on the horizontal axis. Types of Functions 7/9/2013 15 Types of Functions 7/9/2013 15

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**Linear Function Example**

Constant, Linear and Non-Linear x y m = 1 and b = 2 y-intercept : (0,b) = (0,2) Slope : m = 1 x-intercept : (–2, 0) (0, 2) Intercepts and Graphs from Formulas In slope-intercept form the general linear equation is y = mx + b where m is the slope of the graph and b is the y-coordinate of the vertical intercept ( 0, b ). The horizontal intercept can also be read almost immediately from the equation as ( – b/m, 0 ). In this particular case, m = 1 and b = 2. So we know that the graph is a line with slope 1 and that it intersects the y-axis at (0, 2). We could use this information to find another point on the graph, in order to draw the graph. Or, we can find the horizontal intercept as shown and use that point, along with the vertical intercept to draw the graph. The intercepts are convenient because they are easy to locate, always having one coordinate that is 0. Once two distinct points on the graph are established, we simply draw the line through the two points. Note again that the vertical intercept lies on the vertical axis, and the horizontal intercept lies on the horizontal axis. ( ) , b m – = ( ) 2 1 – , = ( ) – 2 0 , Types of Functions 7/9/2013 16 Types of Functions 7/9/2013 16

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**Linear Functions Example**

Constant, Linear and Non-Linear x y Find the equation Intercepts (0,b) = (0, 3) (0, 3) ( ) b m – , = ( ) 4 , 0 (4, 0) b m – = 4 b = 3 and = m b 4 – 3 4 – = Intercepts and Equations from Graphs Here we are given a graph and asked to find the equation of the graph. Knowing that the graph is a line tells us that the general form of the equation can be written as y = mx + b. If we can locate the y-intercept, then we know the coordinate b in the equation. Knowing the form of the x-coordinate for the x-intercept, as shown, allows us to find the slope m. As illustrated, we locate the y-intercept at (0, 3) = (0, b), which yields b = 3. By observing that the x-intercept is (4, 0), and knowing that (4, 0) = (–b/m, 0) we have 4 = –b/m = –3/m so that m = –3/4 and the equation y = mx + b is then y = (–3/4)x + 3 General equation y = x + 3 3 4 – y = mx + b becomes Types of Functions 7/9/2013 17 Types of Functions 7/9/2013 17

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**Constant, Linear and Non-Linear**

Average Rate of Change For function f(x) average rate of change of f(x) over any interval [ x1 , x2 ] is f(x2) – f(x1) x2 – x1 = ∆f ∆x y x y = x2 + 1 NOTE: Nonlinear Functions The function f(x) = x2 + 1 is not linear because it does not have the linear form. However, there is something else that is not the same. A linear function has the property of constant slope, or rate of change. If the value of x is increased by 1, then the linear function increases or decreases by an amount a, where f(x) = ax + b. In this example, if we increase x by 1 at an arbitrary value of x, we have f(x + 1) – f(x) = ( (x + 1)2 + 1) – (x2 + 1) = (x2 +2x + 1) +1 – x2 – 1 = 2x + 1 So, the rate of change depends on where we choose to make the measurement and is therefore not constant. We note that the function is a second degree polynomial, but its rate of change is a linear function. Does the new linear function have a rate of change, and if so, what is it? What interpretation can be given to this second rate of change? Rate not constant … depends x1 and x2 ∆f ∆x So f not linear ! x1 x2 Types of Functions 7/9/2013 18 Types of Functions 7/9/2013 18

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**Constant, Linear and Non-Linear**

Average Rate of Change y x Example: f(x) = x2 + 1 y = x2 + 1 ∆f ∆x (x22 + 1) – (x12 + 1) x2 – x1 = ∆f x22 – x12 x2 – x1 = ∆x x1 x2 (x2 + x1)(x2 – x1) x2 – x1 = Nonlinear Functions The function f(x) = x2 + 1 is not linear because it does not have the linear form. However, there is something else that is not the same. A linear function has the property of constant slope, or rate of change. If the value of x is increased by 1, then the linear function increases or decreases by an amount a, where f(x) = ax + b. In this example, if we increase x by 1 at an arbitrary value of x, we have f(x + 1) – f(x) = ( (x + 1)2 + 1) – (x2 + 1) = (x2 +2x + 1) +1 – x2 – 1 = 2x + 1 So, the rate of change depends on where we choose to make the measurement and is therefore not constant. We note that the function is a second degree polynomial, but its rate of change is a linear function. Does the new linear function have a rate of change, and if so, what is it? What interpretation can be given to this second rate of change? NOTE: x1 + x2 = ∆f ∆x Rate of change, i.e. slope, is not constant … depends on x1 and x2 Types of Functions 7/9/2013 19 Types of Functions 7/9/2013 19

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**Constant, Linear and Non-Linear**

Average Rate of Change y x Example: f(x) = x2 + 1 y = x2 + 1 x1 + x2 = ∆f ∆x ∆f On interval [-2, 3] ∆x ∆x = 5 and 5 ∆f = = ∆f ∆x 5 x1 x2 = 1 Nonlinear Functions The function f(x) = x2 + 1 is not linear because it does not have the linear form. However, there is something else that is not the same. A linear function has the property of constant slope, or rate of change. If the value of x is increased by 1, then the linear function increases or decreases by an amount a, where f(x) = ax + b. In this example, if we increase x by 1 at an arbitrary value of x, we have f(x + 1) – f(x) = ( (x + 1)2 + 1) – (x2 + 1) = (x2 +2x + 1) +1 – x2 – 1 = 2x + 1 So, the rate of change depends on where we choose to make the measurement and is therefore not constant. We note that the function is a second degree polynomial, but its rate of change is a linear function. Does the new linear function have a rate of change, and if so, what is it? What interpretation can be given to this second rate of change? On interval [0, 5] NOTE: ∆x = 5 and 25 ∆f = Rate of change, i.e. slope, is not constant = ∆f ∆x 5 25 = 5 … so f is not linear Types of Functions 7/9/2013 20 Types of Functions 7/9/2013 20

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**Constant, Linear and Non-Linear**

Nonlinear Functions Any function f(x) NOT of form Example 1: f(x) = x2 + 1 f(x) = ax + b x y P2 Average rate of change (slope) not constant … Nonlinear Functions (continued) As before, the function f(x) = x2 + 1 is not linear because it does not have the linear form. And, as before, the average rate of change (or slope) is not constant. There are other types of functions that fall into this category, as illustrated in the slide. They all have the property that the rate of change depends on where we choose to make the measurement and is thus not constant. P3 P4 P1 so function not linear Types of Functions 7/9/2013 21 Types of Functions 7/9/2013 21

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**Constant, Linear and Non-Linear**

Nonlinear Functions Any function f(x) NOT of form Example 2: f(x) = ax + b x y g(x) | | x = Average rate of change (slope) not constant … Nonlinear Functions (continued) As before, the function f(x) = x2 + 1 is not linear because it does not have the linear form. And, as before, the average rate of change (or slope) is not constant. There are other types of functions that fall into this category, as illustrated in the slide. They all have the property that the rate of change depends on where we choose to make the measurement and is thus not constant. P1 P4 P2 P3 so function not linear Types of Functions 7/9/2013 22 Types of Functions 7/9/2013 22

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**Constant, Linear and Non-Linear**

Nonlinear Functions Any function f(x) NOT of form Example 3: f(x) = ax + b x y h(x) = 1 1 – x , x < 1 P4 Nonlinear Functions (continued) As before, the function f(x) = x2 + 1 is not linear because it does not have the linear form. And, as before, the average rate of change (or slope) is not constant. There are other types of functions that fall into this category, as illustrated in the slide. They all have the property that the rate of change depends on where we choose to make the measurement and is thus not constant. Average rate of change (slope) not constant … P2 P1 P3 so function not linear Asymptote Types of Functions 7/9/2013 23 Types of Functions 7/9/2013 23

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**The Difference Quotient**

Constant, Linear and Non-Linear The Difference Quotient Consider a nonlinear function f Secant line through (x, f(x)) and (x + h, f(x + h)) has slope m y = f(x) x Slope m = f x m (x + h, f(x + h)) The Difference Quotient For any function f having a graph, we can ask what the slope of the line through any two points on the graph would be. In the illustration we show the graph of some function f and choose point (x, f(x)) and then increment x by a small amount h. Then we locate the point (x+h, f(x+h)) on the graph and draw the line through these two points. This line is called a secant line and clearly does not coincide with the graph. We find the slope of the secant line, as shown in the illustration. This slope is called the difference quotient for the function over the closed interval [x, x+h]. This slope represents the average rate of change of f(x) over this interval. This is exactly the slope of the secant line through the graph at the endpoints of the interval. An interesting question is: what happens to m as the increment h is decreased and allowed to “approach” 0 ? = f(x + h) – f(x) (x + h) – x f (x, f(x)) x = f(x + h) – f(x) h x x + h h = x Types of Functions 7/9/2013 Types of Functions 7/9/2013 24

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**The Difference Quotient**

Constant, Linear and Non-Linear The Difference Quotient Consider a nonlinear function f Secant line through (x, f(x)) and (x + h, f(x + h)) has slope m y = f(x) x Slope m = f(x + h) – f(x) h m (x + h, f(x + h)) Difference quotient for f on [ x, x+h ] , i.e. the The Difference Quotient For any function f having a graph, we can ask what the slope of the line through any two points on the graph would be. In the illustration we show the graph of some function f and choose point (x, f(x)) and then increment x by a small amount h. Then we locate the point (x+h, f(x+h)) on the graph and draw the line through these two points. This line is called a secant line and clearly does not coincide with the graph. We find the slope of the secant line, as shown in the illustration. This slope is called the difference quotient for the function over the closed interval [x, x+h]. This slope represents the average rate of change of f(x) over this interval. This is exactly the slope of the secant line through the graph at the endpoints of the interval. An interesting question is: what happens to m as the increment h is decreased and allowed to “approach” 0 ? f (x, f(x)) average rate of change of f from x to x + h x x x + h h = x Types of Functions 7/9/2013 25 Types of Functions 7/9/2013 25

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**The Difference Quotient**

Constant, Linear and Non-Linear The Difference Quotient More Secants What happens to m as h gets smaller and smaller? What is the slope of the tangent line at x ? What do the secants approach as m = f(x + h) – f(x) h y = f(x) x Slope m As h m ? , The Difference Quotient For any function f having a graph, we can ask what the slope of the line through any two points on the graph would be. In the illustration we show the graph of some function f and choose point (x, f(x)) and then increment x by a small amount h. Then we locate the point (x+h, f(x+h)) on the graph and draw the line through these two points. This line is called a secant line and clearly does not coincide with the graph. We find the slope of the secant line, as shown in the illustration. This slope is called the difference quotient for the function over the closed interval [x, x+h]. This slope represents the average rate of change of f(x) over this interval. This is exactly the slope of the secant line through the graph at the endpoints of the interval. An interesting question is: what happens to m as the increment h is decreased and allowed to “approach” 0 ? That is, if successively smaller values of h are chosen, the associated secant lines form a sequence of lines all passing through the original point (x+h, f(x+h)) on the graph of f . It becomes clear that the secant lines approach the line tangent to the graph at that point. What is the slope of the tangent line? This question cannot be answered without the concept of a limit, a notion from differential calculus. The reason is that the algebraic method for finding the slopes of the secant lines depends on having two points on the graph through which to draw a secant line. When there is only one point and the line tangent to the graph at that point, algebraic methods fail. Tangent Line x x+h x+h x+h h ? Types of Functions 7/9/2013 26 Types of Functions 7/9/2013 26

27
**The Difference Quotient**

Constant, Linear and Non-Linear The Difference Quotient Example: f(x) = x2 – 4 f(x + h) – f(x) (x + h) – x m = x y = ((x + h)2 – 4) – (x2 – 4) (x + h) – x = x2 + 2xh + h2 – 4 – x2 + 4 h The Difference Quotient: Example For f(x) = x2 – 4 try computing slope using limit of difference quotient at say x = 1 and x = 3. Find lines with these slopes and fit to the curve at x =1 and x = 3. To find the lines, use the point-slope form of the equation – a nice opportunity to introduce this form. What do these lines suggest? Do they appear to be tangent to the curve at these points? Discuss average rate of change and instantaneous rate of change (the slope of the tangent line). This is a good exercise for the graphing calculator. Plot the graph of f(x) = x2 – 4 and then plot the lines with slope 2 and 6 that pass through the points (1, -3) and (3, 5) respectively. The tangency of the lines at these points should be readily apparent. We might ask what the slope of the tangent line is when x = 0. The line is tangent to the graph at the vertex of the parabola. Note that the difference quotient is a precursor to the derivative of differential calculus fame. = 2xh + h2 h = h(2x + h) h m Thus = 2x + h m Types of Functions 7/9/2013 27 Types of Functions 7/9/2013 27

28
**The Difference Quotient**

Constant, Linear and Non-Linear The Difference Quotient Example: f(x) = x2 – 4 2x + h = m 1 x y Clearly, 6 m x as h Slope m depends only on the value of x chosen (3, 5) The Difference Quotient: Example For f(x) = x2 – 4 try computing slope using limit of difference quotient at say x = 1 and x = 3. Find lines with these slopes and fit to the curve at x =1 and x = 3. To find the lines, use the point-slope form of the equation – a nice opportunity to introduce this form. What do these lines suggest? Do they appear to be tangent to the curve at these points? Discuss average rate of change and instantaneous rate of change (the slope of the tangent line). This is a good exercise for the graphing calculator. Plot the graph of f(x) = x2 – 4 and then plot the lines with slope 2 and 6 that pass through the points (1, -3) and (3, 5) respectively. The tangency of the lines at these points should be readily apparent. We might ask what the slope of the tangent line is when x = 0. The line is tangent to the graph at the vertex of the parabola. Note that the difference quotient is a precursor to the derivative of differential calculus fame. 2 At x = 1 , 2x = 2 1 At x = 3 , 2x = 6 (1, –3) Conclusion ? Types of Functions 7/9/2013 28 Types of Functions 7/9/2013 28

29
**Constant, Linear and Non-Linear**

Think about it ! Types of Functions 7/9/2013 29 Types of Functions 7/9/2013 29

30
**Linear Function Example**

Constant, Linear and Non-Linear Types of Functions and Rates of Change Constant & Linear FunctionsTypes of Functions and Rates of Change x y Find intercepts and graph for y = x + 2 Note: y = mx + b m = 1 and b = 2 Slope : 1 Intercepts : (0,b) = (0,2) (–2, 0) (0, 2) Intercepts and Graphs from Formulas In slope-intercept form the general linear equation is y = mx + b where m is the slope of the graph and b is the y-coordinate of the vertical intercept ( 0, b ). The horizontal intercept can also be read almost immediately from the equation as ( – b/m, 0 ). In this particular case, m = 1 and b = 2. So we know that the graph is a line with slope 1 and that it intersects the y-axis at (0, 2). We could use this information to find another point on the graph, in order to draw the graph. Or, we can find the horizontal intercept as shown and use that point, along with the vertical intercept to draw the graph. The intercepts are convenient because they are easy to locate, always having one coordinate that is 0. Once two distinct points on the graph are established, we simply draw the line through the two points. Note again that the vertical intercept lies on the vertical axis, and the horizontal intercept lies on the horizontal axis. ( ) , b m – = ( ) 2 1 – , = ( ) – 2 0 , Types of Functions 7/9/2013 30 Section 1.4 v5.0 Types of FunctionsSection 1.4 v5.0 Types of Functions 2/3/2013 6/5/20132/3/2013 7/9/2013 30 30

31
**The Difference Quotient**

Constant, Linear and Non-Linear Types of Functions and Rates of Change Constant & Linear FunctionsTypes of Functions and Rates of Change The Difference Quotient More Secants What happens to m as h gets smaller and smaller? What is the slope of the tangent line at x ? m = f(x + h) – f(x) h y = f(x) x Slope m As h m ? , The Difference Quotient For any function f having a graph, we can ask what the slope of the line through any two points on the graph would be. In the illustration we show the graph of some function f and choose point (x, f(x)) and then increment x by a small amount h. Then we locate the point (x+h, f(x+h)) on the graph and draw the line through these two points. This line is called a secant line and clearly does not coincide with the graph. We find the slope of the secant line, as shown in the illustration. This slope is called the difference quotient for the function over the closed interval [x, x+h]. This slope represents the average rate of change of f(x) over this interval. This is exactly the slope of the secant line through the graph at the endpoints of the interval. An interesting question is: what happens to m as the increment h is decreased and allowed to “approach” 0 ? That is, if successively smaller values of h are chosen, the associated secant lines form a sequence of lines all passing through the original point (x+h, f(x+h)) on the graph of f . It becomes clear that the secant lines approach the line tangent to the graph at that point. What is the slope of the tangent line? This question cannot be answered without the concept of a limit, a notion from differential calculus. The reason is that the algebraic method for finding the slopes of the secant lines depends on having two points on the graph through which to draw a secant line. When there is only one point and the line tangent to the graph at that point, algebraic methods fail. x x+h x+h x+h Types of Functions 7/9/2013 31 Section 1.4 v5.0 Types of FunctionsSection 1.4 v5.0 Types of Functions 7/9/2013 2/3/2013 6/5/20132/3/2013 31 31

32
**The Difference Quotient**

Types of Functions and Rates of Change Constant, Linear and Non-Linear Constant & Linear FunctionsTypes of Functions and Rates of Change The Difference Quotient More Secants What happens to m as h gets smaller and smaller? What is the slope of the tangent line at x ? What do the secants approach as m = f(x + h) – f(x) h y = f(x) x As h m ? , The Difference Quotient For any function f having a graph, we can ask what the slope of the line through any two points on the graph would be. In the illustration we show the graph of some function f and choose point (x, f(x)) and then increment x by a small amount h. Then we locate the point (x+h, f(x+h)) on the graph and draw the line through these two points. This line is called a secant line and clearly does not coincide with the graph. We find the slope of the secant line, as shown in the illustration. This slope is called the difference quotient for the function over the closed interval [x, x+h]. This slope represents the average rate of change of f(x) over this interval. This is exactly the slope of the secant line through the graph at the endpoints of the interval. An interesting question is: what happens to m as the increment h is decreased and allowed to “approach” 0 ? That is, if successively smaller values of h are chosen, the associated secant lines form a sequence of lines all passing through the original point (x+h, f(x+h)) on the graph of f . It becomes clear that the secant lines approach the line tangent to the graph at that point. What is the slope of the tangent line? This question cannot be answered without the concept of a limit, a notion from differential calculus. The reason is that the algebraic method for finding the slopes of the secant lines depends on having two points on the graph through which to draw a secant line. When there is only one point and the line tangent to the graph at that point, algebraic methods fail. Tangent Line x h ? Types of Functions 7/9/2013 32 Section 1.4 v5.0 Types of Functions Types of FunctionsSection 1.4 v5.0 2/3/2013 6/5/20132/3/2013 7/9/2013 32 32

33
**Constant, Linear and Non-Linear**

Types of Functions Constant & Linear FunctionsTypes of Functions and Rates of Change Constant, Linear and Non-Linear Types of Functions and Rates of Change Linear Function Examples f (x) = 2x + 1 , for x A where A = { -2, -1, 1, 3, 4 } x y x y = f(x) -2 -3 -1 1 3 3 7 4 9 Linear Function Examples: Example 1 The illustration shows an example of the symbolic form of a linear function, defined over a finite domain A, as shown. This means, not only that we can graph the function easily, but also that we can list the function in a table. We show via the arrow connecters how each domain element is matched by the symbolic function form with a corresponding range element to form an ordered pair that is then plotted in the plane as a graph point. Because it is finite, the domain can be listed in a set. Similarly, since the domain can never be smaller than the range, we can also list the finite range. The graph in this case is a discrete graph and, while all its points lie on the line represented by y = 2x + 1, the graph is not the line but the finite set of points extracted from the table. Discrete Graph Question: Is rate of change constant ? Types of Functions 7/9/2013 33 Types of Functions Section 1.4 v5.0 Types of FunctionsSection 1.4 v5.0 2/3/2013 7/9/2013 6/5/20132/3/2013 33 33

34
**Constant, Linear and Non-Linear**

Types of Functions Constant & Linear FunctionsTypes of Functions and Rates of Change Constant, Linear and Non-Linear Types of Functions and Rates of Change Linear Function Examples f (x) = 2x + 1 , for x A where A = { -2, -1, 1, 3, 4 } x y x y = f(x) -2 -3 -1 1 3 3 7 4 9 Linear Function Examples: Example 1 The illustration shows an example of the symbolic form of a linear function, defined over a finite domain A, as shown. This means, not only that we can graph the function easily, but also that we can list the function in a table. We show via the arrow connecters how each domain element is matched by the symbolic function form with a corresponding range element to form an ordered pair that is then plotted in the plane as a graph point. Because it is finite, the domain can be listed in a set. Similarly, since the domain can never be smaller than the range, we can also list the finite range. The graph in this case is a discrete graph and, while all its points lie on the line represented by y = 2x + 1, the graph is not the line but the finite set of points extracted from the table. Discrete Graph Question: Is rate of change constant ? Types of Functions 7/9/2013 34 Types of Functions Section 1.4 v5.0 Types of FunctionsSection 1.4 v5.0 2/3/2013 7/9/2013 6/5/20132/3/2013 34 34

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