2 Solving Radical Equations Method of Solving Radical Equations Step 1: Begin by isolating the radical expression on one side of the equation. If there is more than one radical expression, choose one of the radical expressions to isolate on one side. Step 2: Raise both sides of the equation by the power necessary to “undo” the isolated radical. That is, if the radical is an root, raise both sides to the power.
3 Solving Radical Equations Method of Solving Radical Equations (Cont.) Step 3: If any radical expressions remain, simplify the equation if possible and then repeat steps 1 and 2 until the result is a polynomial equation. When a polynomial equation has been obtained, solve the equation using polynomial methods. Step 4: Check your solutions in the original equation. Any extraneous solutions must be discarded.
4 Example: Solving Radical Equations Solve the radical equation.Note that , so -3 is an extraneous solution.
5 The Cartesian Coordinate System The Cartesian coordinate system consists of two perpendicular real number lines (each of which is an axis). The point of intersection is called the origin of the system, and the four quarters defined by the two lines are called the quadrants of the plane.
6 The Distance FormulaLetting and represent two points on the Cartesian plane, the distance between these two points may be found using the following distance formula, derived from the Pythagorean Theorem:
7 Example: The Distance Formula Determine the distance between and
8 Recognizing Linear Equations in Two Variables Linear Equations in Two Variables A linear equation in two variables, say the variables and , is an equation that can be written in the form where , , and are constants and and are not both zero. This form of such an equation is called the standard form.
9 Intercepts of the Coordinate Axes For an equation in the two variables x and y, it is natural to call the point where the graph crosses the x-axis the x-intercept, and the point where it crosses the y-axis the y-intercept.The y-coordinate of the x-intercept is 0, and the x - coordinate of the y-intercept is 0.x-axisy-axisy-interceptx-intercept
10 Example: InterceptsFind the - and -intercepts of the equation and graph.
11 The Slope of a LineLet stand for a given line in the Cartesian plane, and let and be the coordinates of any two distinct points on . The slope, , of the line, is the ratio which, can be described in words as “change in over change in ” or “rise over run.”
12 Example: Finding Slope Using Two Points Determine the slope of the line passing through the following points.
13 Slope-Intercept Form of a Line If the equation of a non-vertical line in and is solved for , the result is an equation of the form The constant is the slope of the line, and the line crosses the -axis at ; that is, the -intercept of the line is . If the variable does not appear in the equation, the slope is 0 and the equation is simply of the form .
14 Point-Slope Form of a Line Given an ordered pair and a real number an equation for the line passing through the point with slope is Note that , , and are all constants, and that and are variables. Note also that since the line, by definition, has slope , vertical lines cannot be described in this form.
15 The Slopes of Parallel and Perpendicular lines Important Parallel lines have the same slope. For example: Perpendicular lines have slopes that are negative reciprocals of each other. For example:
16 Example: Slopes of Parallel Lines Find the equation, in slope-intercept form, for the line which is parallel to the line and which passes through the pointStep 1: Write equation in slope-intercept form.Use slope to write a new equation that passes through the pointStep 2: Use point-slope form.Step 3: Solve for to obtain slope-intercept form.
17 Solving Linear Inequalities in Two Variables Step 1: Graph the line that results from replacing the inequality symbol with .Solid LineDashed LineNon-strict.Strict.Points on the line included in the solution set.Points on the line excluded from the solution set.
18 Solving Linear Inequalities in Two Variables Select a Test PointSubstitute into the Inequalitytrue statementfalse statementShade entire half-plane that includes the test pointShade entire half-plane that does not include the test point
19 Example: Solving Linear Inequalities Solve the following linear inequality by graphing its solution set.
20 Relations, Domains and Ranges A relation is a set of ordered pairs. Any set of ordered pairs automatically relates the set of first coordinates to the set of second coordinates, and these sets have special names.The domain of a relation is the set of all the first coordinates.The range of a relation is the set of all second coordinates.
21 Functions and the Vertical Line Test The Vertical Line Test If a relation can be graphed in the Cartesian plane, the relation is a function if and only if no vertical line passes through the graph more than once. If even one vertical line intersects the graph of the relation two or more times, the relation fails to be a function.
22 Implied Domain of a Function The domain of the function is implied by the formula used in defining the function. It is assumed that the domain of the function consists of all real numbers at which the function can be evaluated to obtain a real number: any values for the argument that result in division by zero or an even root of a negative number must be excluded from the domain.
23 Quadratic Functions and Their Graphs Vertex Form of a Quadratic Function The graph of the function where a, h and k are real numbers and is a parabola whose vertex is (h,k). The parabola is narrower than if and is broader than . The parabola opens upward if a is positive and downward if a is negative.
24 Quadratic Functions and Their Graphs A quadratic function or a second-degree function of one variable is any function that can be written in the form where a, b, and c are real numbers and The graph of any quadratic function is a roughly U-shaped curve known as a parabola.
25 Commonly Occurring Functions Piecewise-Defined Function A piecewise-defined function is a function defined in terms of two or more formulas, each valid for its own unique portion of the real number line. In evaluating a piecewise-defined function f at a certain value for x, it is important to correctly identify which formula is valid for that particular value.
26 Example: Piecewise-Defined Function Sketch the graph of the functionThe function f is a linear function on the interval and a quadratic function on the interval To graph f, we graph each portion separately, making sure that each formula is applied only on the appropriate interval. The complete graph appears to the right, with the points f(–4) = 6 and f(2) = 4 noted in particular. Also note the use of a closed circle at (–1,0) to emphasize that this point is part of the graph, and the use of an open circle at (–1,1) to indicate that this point is not part of the graph.
27 Example: Variation Problems For the following phrases, write the general formula that applies.“y varies inversely as the n ͭ ͪ power of x”“y is directly proportional to the n ͭ ͪ power of x”“y is inversely proportional to the n ͭ ͪ power of x”d. “y varies directly as the n ͭ ͪ power of x”
28 Shifting, Stretching and Reflecting Graphs Horizontal Shifting Let f(x) be a function whose graph is known, and let h be a fixed real number. If we replace x in the definition of f by x – h, we obtain a new function . The graph of g is the same shape as the graph of f, but shifted to the right by h units if h > 0 and shifted to the left by h units if h < 0.
29 Shifting, Stretching and Reflecting Graphs Vertical Shifting Let f(x) be a function whose graph is known and let k be a fixed real number. The graph of the function is the same shape as the graph of f, but shifted upward if k > 0 and downward if k < 0.
30 Shifting, Stretching and Reflecting Graphs Reflecting With Respect to the AxesLet f(x) be a function whose graph is known.The graph of the function g(x)= –f(x) is the reflection of the graph f with respect to the x-axis.The graph of the function g(x) = f(–x) is the reflection of the graph of f with respect to the y-axis.
31 Shifting, Stretching and Reflecting Graphs Stretching and CompressingLet f(x) be a function whose graph is known, and let a be a positive real number.The graph of the function g(x) = a f(x) is stretched vertically compared to the graph of f if a > 1.The graph of the function g(x) = a f(x) is compressed vertically compared to the graph of f if 0 < a < 1.
32 Shifting, Stretching and Reflecting Graphs Order of Transformations If a function g has been obtained from a simpler function f through a number of transformations, g can usually be understood by looking for the transformations in this order: 1. Horizontal shifts 2. Stretching and compressing 3. Reflections 4. Vertical shifts
33 Symmetry of Functions and Equations y-axis SymmetryThe graph of a function f has y-axis symmetry, or is symmetric with respect to the y-axis, if f(−x) = f(x) for all x in the domain of f. Such functions are called even functions.
34 Symmetry of Functions and Equations Origin SymmetryThe graph of a function f has origin symmetry, or is symmetric with respect to the origin, if f(−x) = −f(x)for all x in the domain of f. Such functions are called odd functions.
35 Example Sketch the graphs of the following relations. Solutions: This relation is actually a function, one that we have already graphed. Note that it is indeed an even function and has y-axis symmetry:
36 Example (cont.)b.We do not quite have the tools yet to graph general polynomial functions, but g(x) = x3 − x can be done. For one thing, g is odd: g(−x) = −g(x) (as you should verify). If we now calculate afew values, such as g(0) = 0,g (1) = 0, and g(2) = 6, and reflect these through the origin, we begin to get a good idea of the shape of g.
37 Combining Functions Arithmetically Addition, Subtraction, Multiplication and Division of Functions22.214.171.124.The domain of each of these new functions consists of the common elements (or the intersection of elements) of the domains of f and g individually.
38 Composing FunctionsComposing Functions Let f and g be two functions. The composition of f and g, denoted , is the function defined by . The domain of consists of all x in the domain of g for which g(x) is in turn in the domain of f. The function is read “f composed with g,” or “f of g.”
39 Example: Composing Functions Given f(x) = x2 and g(x) = x + 5 , find: a. = 112 = 121First, we will find g(6) by replacing x with 6 in g(x).Next, we know that f composed with g can also be written Since we already evaluated g(6), we can insert the answer to get f(11).Continued on the next slide…
40 Example: Composing Functions (cont.) Given f(x) = x2 + 2 and g(x) = x + 5 , find: b. = (x + 5)2 + 2 = x2 + 10x = x2 + 10x + 27Again, we know by definition that