9-1. The Function with Equation y = ax 2. Vocabulary. Graphing y = x 2. Lesson

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1 Chapter 9 Lesson 9-1 The Function with Equation = a BIG IDEA The graph of an quadratic function with equation = a, with a 0, is a parabola with verte at the origin. Vocabular parabola refl ection-smmetric ais of smmetr verte Graphing = The simplest quadratic function has equation =. A table of values for = is given below. Notice the smmetr in the second row of the table. Each value and its opposite have the same square. For eample, 3 and ( 3) are both equal to 9. The bottom row of the table shows that the output of the function is positive for a pair of opposite positive and negative input values. opposite Mental Math If (a, b) is in the nd quadrant, in which quadrant is: a. ( a, b)? b. ( a, b)? c. (a, b)? equal This smmetr can be seen in the graph of the equation = at the right, which is a parabola. Ever positive number is the -coordinate of two points on the graph with opposite -coordinates. For eample, 5 is the -coordinate of the points (5, 5) and ( 5, 5). For this reason, the parabola is its own reflection image over the -ais. For this reason we sa the parabola is reflection-smmetric to the -ais. The -ais is called the ais of smmetr of the parabola. The intersection point of a parabola with its ais of smmetr is called the verte of the parabola. The verte of the graph of = is (0, 0). The function = is of the form = a, with a = 1. You should be able to sketch the graph of an equation of this form Quadratic Equations and Functions

2 Lesson 9-1 Graphing = a All equations of the form = a have similar graphs. Activit Step 1 Use the window 0 0, and 0 0 to graph all three equations on our calculator. Sketch the graphs on a single grid on a separate sheet of paper. a. f () = 3 b. g() = c. h() = 3 Step Evaluate f(), g(), and h(). Step 3 Is (, 1) a point on the graph of h()? Eplain how ou know. Step Use a graphing calculator to make a sketch of the following functions. Use the same window ou used for Step 1. a. j() = 0. b. k() = 0.5 Step 5 Evaluate j(3) and k(3). Step Is ( 3,.9) a point on the graph of j()? Eplain how ou know. Step 7 If a parabola is opening up, what must be true about the value of a in = a? If a parabola is opening down, what must be true about a in = a? Properties of the Graph of = a The graph of = a, where a 0, has the following properties: 1. It is a parabola smmetric to the -ais.. Its verte is (0, 0). 3. If a > 0, the parabola opens up. If a < 0, the parabola opens down. Finding Points on the Graph of = a If ou know the -coordinate of a point on the graph of a parabola and the equation, ou can find the -coordinate or coordinates. We illustrate this with a different parabola. GUIDED Eample 1 Consider the following situation: You know the area of a circle and want to fi nd its radius. (continued on net page) The Function with Equation = a 57

3 Chapter 9 Solution Sketch a graph of the familiar formula A =πr, where A is the area of a circle with radius r. Step 1 Make a scale on each ais. In doing this, ask ourself: What are the possible values of r? What are the possible values of A? Step Make a table of values for A when r = 1,, 3, and. Estimate each to the nearest hundredth. The fi rst value has been done for ou. Step 3 Graph the points (r, A) from the table. Step Put an open circle at (0, 0) because 0 is not in the domain of r. Connect (0, 0) and the other points with a curve like a parabola. Step 5 Use our graph to estimate the radius of a circle whose area is 1 square units. r A 1 3.1? 3?? Eample The graph of f() = 1.5 is shown at the right. Estimate if f () =. Solution Draw the horizontal line =. The graph intersects this line at two points. The -coordinates of these points are approimatel.5 and.5. In the net lesson, ou will see how to obtain the eact values of with f () =. Questions COVERING THE IDEAS In 1 and, an equation of a function is given. a. Make a table of and for integer values of from to. b. Graph the equation. c. Tell whether the graph opens up or down. 1. g() = 1. f () = 1 3. Refer to the parabola at the right. a. Does the parabola open up or down? b. The parabola is the graph of a function. Which does the function have, a maimum value or a minimum value? c. Give the coordinates of the verte. d. Give an equation of the ais of smmetr of the parabola.. How are the graphs of = 7 and = 7 related to each other? = 5 Quadratic Equations and Functions

4 Lesson Match each table with the graph it most accuratel represents. a. b. c i. ii. iii Match each graph at the right with one of the equations below. a. = iii b. = 0.5 c. = 3 7. Fill in the Blanks Consider the graph of the function f () = a. a. If a is positive, the graph is a parabola that opens?. b. If a is negative, the graph is a parabola that opens?. i ii. Use the graph of A = πr to estimate the radius of a circle whose area is 0 square units. 9. Consider the graph of the function defined b = 5. a. Without plotting an points, sketch what ou think the graph of this function looks like. b. Make a table of values satisfing this function. Use =, 1.5, 1, 0.5, 0, 0.5, 1, 1.5, and. c. Draw a graph of this function from our table. d. From the graph, estimate the values of for which = 1. The Function with Equation = a 59

5 Chapter 9. Consider the formula A = s for the area A of a square with a side of length s. a. Graph all possible values of s and A on a coordinate plane. b. Eplain how the graph in Part a is like and unlike the graph of = at the start of this lesson. 11. The parabola at the right has equation = 5. a. Find if = 0. b. Find if = 5. c. Find if = APPLYING THE MATHEMATICS 1. Refer to the parabola at the right. Points P and Q are reflection images of each other over the -ais. What are the coordinates of Q? In 13 and 1, fill in the blanks with negative, zero, or positive. 13. Fill in the Blanks Consider the epression 1. a. If is negative, 1 is?. b. If is zero, 1 is?. c. If is positive, 1 is?. d. What do Parts a c tell ou about the graph of = 1? P ( 3, 0) Q 1. Fill in the Blanks Consider the epression. a. If is negative, is?. b. If is zero, is?. c. If is positive, is?. d. What do Parts a c tell ou about the graph of =? 15. What is the onl real number whose square is not a positive number? In 1 and 17, a graph of a function f with f () = a is shown. Find the value of a (3, ) (, ) 530 Quadratic Equations and Functions

6 Lesson Fill in the Blank If a = 0, the graph of the function = a is?. 19. Consider the equation d = 1t, which gives the distance d in feet that an object dropped at time t = 0 will have fallen after t seconds. a. If an object falls 00 feet in t seconds, find t. b. Estimate how long it will take an object to fall 00 feet. REVIEW 0. Skill Sequence Simplif each epression. (Lesson -7) a. 3 7 b. 3 7 c Suppose a bo has sides of length inches, inches, and 1 inches. Find the length of the longest thin pole, like the one shown at the right, which can fit inside the bo. (Hint: First find the diagonal of the base.) (Lesson -) 1 in.. Suppose that t ears ago, Kendra deposited P dollars into a savings account with an in. annual ield of 3%. If she has not deposited or withdrawn an additional mone and the account now contains $500, write an equation involving t and P. (Lesson 7-1) 3. Derek is a tennis instructor at his local gm. He gives lessons to 3 people twice a week. If he charges $5 per person for a lesson, and he works for 15 straight weeks, how much mone will Derek earn? (Lesson 5-) in. The United States Tennis Association is the largest tennis organization in the world with more than 5,000 individual members and 7,000 organizational members. Source: USTA EXPLORATION. Draw a set of aes on graph paper. Aim a lit flashlight at the origin up the -ais. What is the shape of the lit region? Keep the lit end of the flashlight over the origin but tilt the flashlight to raise its bottom. How does the shape of the lit region change? The Function with Equation = a 531