Properties of Graphs of Quadratic Functions

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1 Properties of Graphs of Quadratic Functions y = ax 2 + bx + c 1) For a quadratic function given in standard form a tells us: c is the: 2) Given the equation, state the y-intercept and circle the direction of opening. y = 3x 2 x 8 y = 2 + 3x x 2 y = (x 2) 2 3 Y int = Y int = Y int = Opens: UP or DOWN Opens: UP or DOWN Opens: UP or DOWN 3) Given the following, circle those relations that are quadratic. y = 3x + 5 y = 4x y = 2(x + 2) ) Given the following graph, answer the questions listed below: y a) State the x-intercept(s) b) State the y-intercept c) State the coordinates of the vertex x d) State the equation of the axis of symmetry e) State the domain f) State the range 6 7 y = x 2 + x 6

2 4) Graph the following quadratic function by creating a table of values. Then answer the following questions. y = x 2 x + 2 x y a) State the x-intercept(s) b) State the y-intercept c) State the coordinates of the vertex d) State the equation of the axis of symmetry e) State the domain f) State the range

3 Vertex Form of a Quadratic Function y = a(x h) 2 + k 1) For a Quadratic Function given in vertex form a tells us: h tells us: k tells us: 2) For the following quadratic functions given in vertex form answer the questions listed. y = 1 2 (x 3)2 + 6 y = 5(x + 6) a) State the direction of opening. b) State the coordinates of the vertex. c) State the equation of the axis of symmetry. d) Sketch a graph e) State the domain and range Domain: Range: Domain: Range: f) Does the function have a maximum of minimum value? g) Predict the number of x- intercepts (0, 1, 2)

4 3) Write the equation in vertex and standard form for the following quadratic functions given the vertex and another point. a) Vertex = (6, 4) and Point = (8, 6) b) Find the information from the graph. Vertex Form: Standard Form: Vertex Form: Standard Form:

5 Solving Quadratic Equations by Graphing 1) For each graph, determine the roots of the corresponding quadratic equation. Roots = Roots = Roots = 2) Solve each equation by graphing the corresponding function and determining the zeros (roots). x 2 4x + 5 = 0 x 2 + 2x 1 = 0 Roots = Roots =

6 3) The height, in metres, of a fireworks rocket is modelled by the function h(t) = 4.9t t + 4, where t is the time in seconds after the rocket is fired. The graph of this function is on the left. a) If you are sitting safely at a height of 19 m above the ground at what approximate time(s) is the fireworks rocket at your height? b) How long does it take the firework rocket to hit the ground? c)how high is the fireworks rocket able to go? 4) A football is thrown down field but it is an incomplete pass and hits the ground. The path of the ball is modelled by the function h(d) = 0.01(d 18) Using DESMOS, graph the function and answer the following questions: a) How far from the quarterback was the ball when it hit the ground? b) What was the maximum height of the ball? c) At which height was the ball thrown from? d) If someone was standing 28 m from the quarterback would it be possible for them to intercept the ball? e) If someone was standing 39 m from the quarterback would it be possible for them to intercept the ball?

7 Solving Quadratic Equations by Factoring Hint: place each equation in standard form, factor, use the zero product principle, and solve for x. 1) Solve the following quadratic equations by factoring. a) x 2 + 8x + 15 = 0 b) 6x x = 4 c) 4x 2 = 121 d) 24x + 16x = 0 e) 5x x = 0

8 2) Determine an equation in factored form and in standard form of a parabola that has x intercepts of -2 and 1 that passes through the y axis at -4. Factored Form: Standard Form: 3) Determine an equation in factored form and in standard form for a quadratic function with the following graph: Factored Form: Standard Form:

9 Using the Quadratic Formula to Solve Quadratic Equations x = b ± b2 4ac 2a 1) Solve the following quadratic equations using the quadratic formula. Round to the nearest hundredth where applicable. a) 2x 2 7x + 5 = 0 b) x 2 + 5x 2 = 0

10 c) x 2 2x + 3 = 0 d) 4x = 12x

11 Solving Word Problems 1. The square of a number plus the number itself is What is the number? 2. The length of a rectangle is 2 cm more than the width. The area of the rectangle is 49 cm². Find the width of the rectangle to the nearest tenth of a centimeter. 3. A square is converted into a rectangle by adding 5 cm to its length. The area of the rectangle is 84 cm 2. What are the dimensions of the new shape?

12 4. A landscaper is designing a 6 m by 8 m rectangular garden that will then be surrounded by a uniform border of crushed stone. She has enough crushed stone to cover 72 m 2. What is the width of the border if she uses all of the crushed stone? Use DESMOS for the following questions: 5. A batted baseball is modelled by the function h(d) = d d + 1 where h(t) is the height of the ball in meters and t is the time in seconds, since the ball was hit. If the ball was a line drive over first base (Fair ball) and the fence is 70 m from home plate, will the ball be knocked out of the park if the fence is 2 m high? Justify your answer. 6. Mallory can throw a ball upward at 15 m/s, so that its height, h(t) metres, is given approximately by 2 the function h( t) 5t 20t 2 where t is the time in seconds. Can Mallory throw the ball onto the roof of a building that is 10 m tall? Explain. 7. An object thrown upwards with an initial velocity of 20 m/s from a height of 180 m above the 2 ground can be represented by the equation h( t) 5t 20t 180 where h(t) represents the height of the object in metres and t represents the time (seconds) after the object is thrown. At what time will the object hit the ground?