# CH 9. Quadratic Equations and Functions

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1 9.1: Graph 9.2: Graph 9.3: Solve Quadratic Equations by Graphing 9.4: Use Square Roots to Solve Quadratic Equations 9.5: Solve Quadratic Equations by Completing the Square 9.6: Solve Quadratic Equations by Quadratic Formula 9.7: Interpret the Discriminant 9.8: Compare Linear, Exponential, and Quadratic Models Prerequisite Skills 1. The x-coordinate of a point where a graph crosses the x-axis is a(n). 2. A(n) is a function of the form where, and. 3. Evaluate the expression. a. b. c. KEY VOCABULARY Quadratic function Vertex Quadratic equation Parabola Axis of symmetry Completing the square Parent quadratic function Minimum value Quadratic formula Maximum value Discriminant Page 1

2 9.1 Graph A quadratic function is a nonlinear function that can be written in the standard form where. Every quadratic function has a U-shaped graph called a parabola. In this lesson, you will graph quadratic function where. Parent Quadratic Function The most basic quadratic function in the family of quadratic functions, called the parent quadratic function, is. The graph of is shown below. The lowest or highest point on a parabola is the vertex. The vertex of the graph of is (0, 0) The line that passes through the vertex and divides the parabola into two symmetric parts is called the axis of symmetry. The axis of symmetry for the graph of is the y-axis,. Graph STEP 1 Make a table of values for. x STEP 2 y Plot the points from the table. STEP 3 Draw the smooth curve through the points STEP 4 Compare the graph of and. Both graphs open up and have the same vertex, (0, 0), and axis of symmetry,. The graph of is narrower than the graph of because the graph of is vertical stretch (by factor of 3) of the graph of. Graph STEP 1 Make a table of values for. x y 0 STEP 2 Plot the points from the table. STEP 3 Draw the smooth curve through the points. STEP 4 Compare the graph of and. Both graphs have the same vertex, (0, 0), and axis of symmetry,. However, the graph of is wider than the graph of and it opens down. This is because the graph of is vertical shrink (by factor of ) with a reflection in the x-axis of the graph of. Page 2

3 GRAPHING QUADRATIC FUNCTIONS Example 1 and 2 suggest the following general result: a parabola opens up when the coefficient of is positive and opens down when the coefficient of is negative. Graph STEP 1 Make a table of values for. x STEP 2 y Plot the points from the table. STEP 3 Draw the smooth curve through the points STEP 4 Compare the graph of and. Both graphs open up and have the same axis of symmetry,. However, the vertex of the graph of, (0, 5) is different than the graph of, (0, 0), because the graph of is vertical translation (of 5 units up) of the graph of. Ex) Graph the function. Compare the graph with the graph of. Page 3

4 Graph STEP 1 Make a table of values for. x y 4 4 STEP 2 STEP 3 Plot the points from the table. Draw the smooth curve through the points STEP 4 Compare the graph of and. Both graphs open up and have the same axis of symmetry,. However, the graph of is wider and has a lower vertex than the graph of, because the graph of is vertical shrink and a vertical translation of the graph of. Ex) Graph the function. Compare the graph with the graph of. Page 4

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6 Ex) A solar though has a reflective parabolic surface that is used to collect solar energy. The sun s rays are reflected from the surface toward a pipe that carries water. The heated water produces steam that is used to produce electricity. The graph of the function models the cross section of the reflective surface where x and y are measured in meter. Use the graph to find the domain and range of the function in this situation. Ex) A cross section of the parabolic surface of the antenna shown can be modeled by the graph of the function where x and y are measured in meters. a. Find the domain and range of the function in this situation. Page 6

7 9.2 Graph You can use the properties below to graph any quadratic function. Properties of the Graph of Quadratic Function The graph of is a parabola that: Opens up if and opens down if. Is narrower than the graph of if and wider if. Has an axis of symmetry of. Has a vertex with an x-coordinate of. Has a y-intercept of c. so, the point (0, c) is on the parabola. Ex) Find the axis of symmetry and the vertex, from the given function. Graph STEP 1 Determine whether the parabola opens up or down. Because a > 0, the parabola opens up. STEP 2 STEP 3 Find and draw the axis of symmetry: Find and plot the vertex. The x-coordinate, substitute 1 for x in the function and simplify. ( ) ( ) So, the vertex is ( ) ( ). STEP 4 Plot two points. Choose two x-values less than the x-coordinate of the vertex. Then find the corresponding y-values. STEP 5 STEP 6 x 0 y 2 11 Reflect the points plotted in Step 4, in the axis of symmetry. Draw a parabola through the plotted points. Page 7

8 Minimum and Maximum Values For, the y-coordinate of the vertex is the minimum value of the function if or the maximum value of the function if. Ex) Tell whether the given function has a minimum value or a maximum value. Then find the minimum or maximum value. ( ) ( ) Page 8

9 9.3 Solve Quadratic Equations By Graphing A quadratic equation is an equation that can be written in the standard form where. In Chapter 8, you used factoring to solve a quadratic equation. You can also use graphing to solve a quadratic equation. Notice that the solutions of the equation are the x-intercepts of the graph of the related function. Solve by Factoring Solve by Graphing ( )( ) To solve, graph. From the graph you can see that the x-intercepts are 1 and 5. To solve a quadratic equation by graphing, first write the equation in standard form,. Then graph the related function. The x-intercepts of the graph are the solutions, or roots, of. Number of Solutions of a Quadratic Equation A quadratic equation has two solutions if the graph of its related function has two x-intercepts. A quadratic equation has one solution if the graph of its related function has one x-intercept. A quadratic equation has no real solution if the graph of its related function has no x- intercepts. Ex) Find the solutions of given equation (graph). Page 9

10 9.4 Use Square Roots to Solve Quadratic Equations To use square roots to solve a quadratic equation of the form, first isolate on one side to obtain. Then use the following information about the solutions of to solve the equation. Solve by Taking Square Roots If d > 0, then has two solutions: If d = 0, then has one solution: If d < 0, then has no solution. Ex) Solve quadratic equations a. b. c. d. e. f. g. SIMPLIFYING SQUARE ROOTS In cases where you need to take the square root of a fraction whose numerator and denominator are perfect squares, the radical can be written as a fraction. For example, can be written as because ( ). Ex) Solve a quadratic equation. a. ( ) b. ( ) c. ( ) Page 10

11 9.5 Solve Quadratic Equations by Completing the Square For an expression of the form, you can add a constant c to the expression so that the expression is a perfect square trinomial. This process is called completing the square COMPLETING THE SQUARE Words To complete the square for the expression, add the square of half the coefficient of the term bx. Algebra ( ) = ( ) Ex) Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. 1) 2) 3) 4) SOLVING EQUATIONS The method of completing the square can be used to solve any quadratic equation. To use completing the square to solve a quadratic equation, you must write the equation in the form. Ex) Solve the equation by completing the square. 1) Page 11

12 2) 3) 4) 5) Word Problem You decide to use chalkboard paint to create a chalkboard on a door. You want the chalkboard to have a uniform border as shown. You have enough chalkboard paint to cover 6 square feet. Find the width of the border to the nearest inch. Page 12

13 9.6 Solve Quadratic Equations by the Quadratic Formula By completing the square for the quadratic equation, you can develop a formula that gives the solutions of any quadratic equation in standard form. This formula is called the quadratic formula. The Quadratic Formula The solutions of the quadratic equation are where and. Ex) Solve the equation. 1) 2) 3) 4) 5) Page 13

14 CONCEPT SUMMARY Methods for Solving Quadratic Equations Method Lesson(s) When to Use Factoring 9.4~9.8 Use when a quadratic equation can be factored easily. Graphing 10.3 Use when approximate solutions are adequate. Finding square roots Completing the square Use when solving an equation that can be written in the form. Can be used for any quadratic equation but is simplest to apply when and b is an even number. Quadratic formula 10.6 Can be used for any quadratic equation. Ex) Tell what method you would use to solve the quadratic equation. Explain your choice(s). 1) 2) 3) 4) 5) 6) Page 14

15 In the quadratic formula, the expression associated equation. is called the discriminant of the Discriminant Because the discriminant is under the radical symbol, the value of the discriminant can be used to determine the number of solutions of a quadratic equation and the number of x- intercepts of the graph of the related function. KEY CONCEPT Using the Discriminant of Value of the discriminant Number of solutions Two solutions One solution No solution Graph of Two x-intercepts One x-intercept No x-intercept Ex) Tell whether the equation has two solutions, one solution, or no solution. 1) 2) 3) Page 15

16 4) 5) 6) 7) Ex) Find the number of x-intercepts of the graph of the function. 1) 2) 3) 4) Page 16

17 9.7 Solve Systems with Quadratic Equations You have solved systems of linear equations using the graph-and-check method and using the substitution method. You can use both of these techniques to solve a system of equations involving nonlinear equations, such as quadratic equations. Recall that the substitution method consists of the following three steps. STEP 1 STEP 2 STEP 3 Solve one of the equations for one of its variables. Substitute the expression from Step 1 into the other equation and solve for the other variable. Substitute the value from Step 2 into one of the original equations and solve. POINT OF INTERSECTION When you graph a system of equations, the graphs intersect at each solution of the system. For a system consisting of a linear equation and a quadratic equation the number of intersections, and therefore solutions, can be zero, one, or two. KEY CONCEPT Systems With One Linear Equation and One Quadratic Equation There are three possibilities for the number of points of intersection No Solution One Solution Two Solution Page 17

18 Ex) Use the substitution method to solve the system 1) 2) 3) 4) Page 18

19 9.8 Compare Linear, Exponential, and Quadratic Models So far you have studied linear functions, exponential functions, and quadratic functions. You can use these functions to model data. KEY CONCEPT Linear, Exponential, and Quadratic Functions Linear Function Exponential Function Quadratic Function Ex) Use a graph to tell whether the ordered pairs represent a linear function, an exponential function, or a quadratic function. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Page 19

20 ( ) ( ) ( ) ( ) ( ) DIFFERENCES AND RATIOS A table of values represents a linear function if the differences of successive y-values are all equal. A table of values represents an exponential function if the ratios of successive y-values are all equal. In both cases, the increments between successive x-values need to be equal. Use differences or ratios to tell whether the table of values represents a linear function, an exponential function, or a quadratic function. Extend the table to find the y-value for the next x-value. 1) x y Page 20

21 2) x y WRITING AN EQUATION When you decide that a set of ordered pairs represents a linear, an exponential, or a quadratic function, you can write an equation for the function. In this lesson, when you write an equation for a quadratic function, the equation will have the form Ex) Tell whether the table of values represents a linear function, an exponential function, or a quadratic function. Then write an equation for the function. 1) x y ) x y ) x y Page 21

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