Chapter 2 Functions and Graphs

2.1 Basics of Functions & Their Graphs

Objectives Find the domain & range of a relation. Determine whether a relations is a function. Determine is an equation represents a function. Evaluate a function. Graph functions by plotting points. Use the vertical line test to identify functions. Obtain information from a graph. Identify the domain & range from a graph. Identify intercepts from a graph.

Domain & Range A relation is a set of ordered pairs. Domain: first components in the relation (independent) Range: second components in the relation (dependent, the value depends on what the domain value is) Functions are SPECIAL relations: A domain element corresponds to exactly ONE range element.

EXAMPLE Consider the function: eye color (assume all people have only one color, and it is not changeable) It IS a function because when asked the eye color of each person, there is only one answer. i.e. {(Joe, brown), (Mo, blue), (Mary, green), (Ava, brown), (Natalie, blue)} NOTE: the range values are not necessarily unique.

Evaluating a function Common notation: f(x) = function Evaluate the function at various values of x, represented as: f(a), f(b), etc. Example: f(x) = 3x – 7 f(2) = 3(2) – 7 = 6 – 7 = -1 f(3 – x) = 3(3 – x) – 7 = 9 – 3x – 7 = 2 – 3x

Graphing a functions Horizontal axis: x values Vertical axis: y values Plot points individually or use a graphing utility (calculator or computer algebra system) Example:

Table of function values X (domain) Y (range)

Graphs of functions

Can you identify domain & range from the graph? Look horizontally. What all x-values are contained in the graph? That’s your domain! Look vertically. What all y-values are contained in the graph? That’s your range!

What is the domain & range of the function with this graph?

Can you determine if a relation is a function or NOT from the graph? Recall what it means to be a function: an x-value is assigned ONLY one y-value (it need not be unique). So, on the graph, if the x value is paired with MORE than one y value there would be two points directly on a vertical line. THUS, the vertical line test! If 2 or more points fall on a vertical line that would cross any portion of your graph, it is NOT the graph of a function.

Finding intercepts: X-intercept: where the function crosses the x-axis. What is true of every point on the x-axis? The y-value is ALWAYS zero. Y-intercept: where the function crosses the y-axis. What is true of every point on the y-axis? The x-value is ALWAYS zero. Can the x-intercept and the y-intercept ever be the same point? YES, if the function crosses through the origin!

2.2 More of Functions and Their Graphs

Objectives Find & simplify a function’s difference quotient. Understand & use piecewise functions. Identify intervals on which a function increases, decreases, or is constant. Use graphs to locate relative maxima or minima. Identify even or odd functions & recognize the symmetries. Graph step functions.

Difference Quotient Useful in discussing the rate of change of function over a period of time EXTREMELY important in calculus, (h represents the difference in two x values)

Find the difference quotient

What is a piecewise function? A function that is defined differently for different parts of the domain. Examples: You are paid $10/hr for work up to 40 hrs/wk and then time and a half for overtime.

Increasing and Decreasing Functions Increasing: Graph goes “up” as you move from left to right. Decreasing: Graph goes “down” as you move from left to right. Constant: Graph remains horizontal as you move from left to right.

Even & Odd Functions Even functions are those that are mirrored through the y-axis. (if –x replaces x, the y value remains the same) (i.e. 1 st quadrant reflects into the 2 nd quadrant) Odd functions are those that are mirrored through the origin. (if –x replaces x, the y value becomes –y) (i.e. 1 st quadrant reflects into the 3 rd quadrant)

Determine if the function is even, odd, or neither. 1.Even 2.Odd 3.Neither

2.3 Linear Functions & Slope

Objectives Calculate a line’s slope. Write point-slope form of a line’s equation. Write & graph slope-intercept from of a line’s equation. Graph horizontal or vertical lines. Recognize & use the general form of a line’s equation. Use intercepts to graph. Model data with linear functions and predict.

What is slope? The steepness of the graph, the rate at which the y values are changing in relation to the changes in x. How do we calculate it?

A line has one slope Between any 2 pts. on the line, the slope MUST be the same. Use this to develop the point-slope form of the equation of the line. Now, you can develop the equation of any line if you know either a) 2 points on the line or b) one point and the slope.

Find the equation of the line that goes through (2,5) and (-3,4) 1 st : Find slope of the line m= 2 nd : Use either point to find the equation of the line & solve for y.

Slope-Intercept Form of the Equation of the Line It is often useful to express the line in slope-intercept form, meaning that the equation quickly reveals the slope of the line and where it intercepts the y-axis. It is REALLY use of the point-slope form, except that the point is the intercept, (0,b). y - b=m(x - 0) becomes y = mx + b This creates a quick equation to graph.

Horizontal & Vertical lines What is the slope of a vertical line? –It is INFINITELY steep, it only rises. It is SO steep, we can’t define it, therefore undefined slope. –Look at points on a line, i.e. (-4,8),(-4,9),(-4,13),(-4,0). We don’t care what value y has, all that matters is that x= - 4. Therefore, that is the equation of the line! What is the slope of a horizontal line? –There is no rise, it only runs, the change in y is zero, so the slope = 0. –Look at points on the line, i.e., (2,5), (-2,5), (17,5), etc. We don’t care what value x has, all that matters is that y = 5. Therefore, that is the equation of the line!

2.4 More on SLOPE Objectives –Find slopes & equations of parallel & perpendicular lines –Interpret slope as a rate of change –Find a function’s average rate of change

Parallel lines Slopes are equal. If you are told a line is parallel to a given line, you automatically know the slope of your new line (same as the given!). Find the equation of the line parallel to y=2x-7 passing through the point (3,-5). –slope = 2, passes through (3,-5) y - (-5) = 2(x – 3) y + 5 = 2x – 6 y = 2x - 11

Perpendicular lines Lines meet to form a right angle. If one line has a very steep negative slope, in order to form a right angle, it must intersect another line with a gradual positive slope. The 2 lines graphed here illustrate that relationship.

What about the intersection of a horizontal line and a vertical line? They ALWAYS intersect at a right angle. Since horizontal & vertical lines are neither positive or negative, we simply state that they are, indeed, ALWAYS perpendicular. What about all other lines? In order to be perpendicular, their slopes must be the negative reciprocal of each other. (HINT: think about the very steep negative-sloped line perpendicular to the gradual positive-sloped line)

Find the equation of the line perpendicular to y = 2x – 7 through the point (2,7). 1.y = ½ x y = - 2 x + 1/7 3.y = - ½ x + 4/3 4.y = - ½ x + 8

Average Rate of Change Slope thus far has referred to the change of y as related to the change in x for a LINE. Can we have slope of a nonlinear function? We CAN talk about the slope between any 2 points on the curve – this is the average rate of change between those 2 points!

2.5 Transformation of Functions Recognize graphs of common functions Use vertical shifts to graph functions Use horizontal shifts to graph functions Use reflections to graph functions Use vertical stretching & shrinking to graph functions Use horizontal stretching & shrinking to graph functions Graph functions w/ sequence of transformations

Vertical shifts –Moves the graph up or down –Impacts only the “y” values of the function –No changes are made to the “x” values Horizontal shifts –Moves the graph left or right –Impacts only the “x” values of the function –No changes are made to the “y” values

Recognizing the shift from the equation, examples of shifting the function f(x) = Vertical shift of 3 units up Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

Combining a vertical & horizontal shift Example of function that is shifted down 4 units and right 6 units from the original function.

Reflecting Across x-axis (y becomes negative, -f(x)) Across y-axis (x becomes negative, f(-x))

Horizontal stretch & shrink We’re MULTIPLYING by an integer (not 1 or 0). x’s do the opposite of what we think they should. (If you see 3x in the equation where it used to be an x, you DIVIDE all x’s by 3, thus it’s compressed horizontally.)

VERTICAL STRETCH (SHRINK) y’s do what we think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high or low!)

Sequence of transformations Follow order of operations. Select two points (or more) from the original function and move that point one step at a time. f(x) contains (-1,-1), (0,0), (1,1) 1 st transformation would be (x+2), which moves the function left 2 units (subtract 2 from each x), pts. are now (-3,-1), (-2,0), (- 1,1) 2 nd transformation would be 3 times all the y’s, pts. are now (- 3,-1), (-2,0), (-1,3) 3 rd transformation would be subtract 1 from all y’s, pts. are now (-3,-2), (-2,-1), (-1,2)

Graph of Example

2.6 Combinations of Functions; Composite Functions Objectives –Find the domain of a function –Combine functions using algebra. –Form composite functions. –Determine domains for composite functions. –Write functions as compositions.

Using basic algebraic functions, what limitations are there when working with real numbers? A) You canNOT divide by zero. –Any values that would result in a zero denominator are NOT allowed, therefore the domain of the function (possible x values) would be limited. B) You canNOT take the square root (or any even root) of a negative number. Any values that would result in negatives under an even radical (such as square roots) result in a domain restriction.

Example Find the domain There are x’s under an even radical AND x’s in the denominator, so we must consider both of these as possible limitations to our domain.

Algebra of functions (f + g)(x) = f(x) + g(x) (f - g)(x) = f(x) – g(x) (fg)(x) = f(x)g(x)

Composition of functions Composition of functions means the output from the inner function becomes the input of the outer function. f(g(3)) means you evaluate function g at x=3, then plug that value into function f in place of the x. Notation for composition:

2.7 Inverse Functions Objectives –Verify inverse functions –Find the inverse of a function. –Use the horizontal line test to deterimine one- to-one. –Given a graph, graph the inverse. –Find the inverse of a function & graph both functions simultaneously.

What is an inverse function? A function that “undoes” the original function. A function “wraps an x” and the inverse would “unwrap the x” resulting in x when the 2 functions are composed on each other.

How do their graphs compare? The graph of a function and its inverse always mirror each other through the line y=x. Example:y = (1/3)x + 2 and its inverse = 3(x-2) Every point on the graph (x,y) exists on the inverse as (y,x) (i.e. if (-6,0) is on the graph, (0,-6) is on its inverse.

Do all functions have inverses? Yes, and no. Yes, they all will have inverses, BUT we are only interested in the inverses if they ARE A FUNCTION. DO ALL FUNCTIONS HAVE INVERSES THAT ARE FUNCTIONS? NO. Recall, functions must pass the vertical line test when graphed. If the inverse is to pass the vertical line test, the original function must pass the HORIZONTAL line test (be one-to-one)!

How do you find an inverse? “Undo” the function. Replace the x with y and solve for y.

2.8 Distance & Midpoint Formulas; Circles Objectives –Find distance between 2 points. –Find midpoint of a line segment. –Write standard form of a circle’s equation. –Give center & radius of a circle whose equation is in standard form. –Convert general form of a circle’s equation to standard form.

Distance formula is an application of the Pythagorean theorem. Recall for any right triangle, the sum of the squares of the legs equals the square of the hypotenuse. For any 2 points, you can imagine a rt. Triangle that would have those 2 pts. The length of the vertical leg is the difference in the y values. The length of the horizontal leg is the difference between the x’s. Distance between the points=

Midpoint is the point in the middle, therefore the x-value is the average of the 2 given x’s and the y-value is the average of the 2 given y’s. Midpoint of

Standard form or a circle with radius=r and center at (h,k)

A circle centered at (-2,5) with a radius=7 has what equation? A circle whose equation is has what as its center and radius? center = (2,-6) and radius=2

What if the equation is not in standard form? You may have to complete the square.