Chapter 6: Quadratic Functions

Transcription

1 Chapter 6: Quadratic Functions Section 6.1 Chapter 6: Quadratic Functions Section 6.1 Exploring Quadratic Relations Terminology: Quadratic Relations: A relation that can be written in the standard form y = ax 2 + bx + c, where a 0; A function is considered a quadratic function if its highest power when expanded is of degree 2 (ie a squared term) Ex. y = 4x 2 + 2x + 1 Parabola: The shape of the graph of any quadratic relation. (Parabolas are -shape or -shape) Identifying a Quadratic Function from an Equation To determine if a function is quadratic, you must identify if the function has a highest power of 2 when expanded. Any function that does not have a degree of 2, is not quadratic. Ex. Which functions are quadratic? (a) y = 2x 3 (b) y = 5 2x 2 + 3x (c) y = (x + 2)(x 3) (d) y = x 3 2x 2 + 3x 5 (e) y = 3x(x + 7) Features of a Quadratic Functions in Standard Form Given the quadratic function in standard form: y = 2x 2 + 5x + 3 Will produce the graph shown to the right: What do you notice about the shape of the graph?

2 Chapter 6: Quadratic Functions Section 6.1 Exploring the Effects of the a value: If we change the a-value of the equation, how will this impact the graph in each situation? Original Function New Function How has it Changed? y = 2x 2 + 5x + 3 y = 2x 2 + 5x + 3 y = 2x 2 + 5x + 3 y = 1 2 x2 + 5x + 3 y = 2x 2 + 5x + 3 y = 3x 2 + 5x + 3 y = 2x 2 + 5x + 3 y = 3 4 x2 + 5x + 3

3 Chapter 6: Quadratic Functions Section 6.1 Exploring the Effects of the c value: If we change the c-value of the equation, how will this impact the graph in each situation? Original Function New Function How has it Changed? y = 2x 2 + 5x + 3 y = 2x 2 + 5x + 6 y = 2x 2 + 5x + 3 y = 2x 2 + 5x 4 y = 2x 2 + 5x + 3 y = 2x 2 + 5x

4 Chapter 6: Quadratic Functions Section 6.1 Exploring the Effects of the b value: If we change the b-value of the equation, how will this impact the graph in each situation? Original Function New Function How has it Changed? y = 2x 2 + 5x + 3 y = 2x 2 + 9x + 3 y = 2x 2 + 5x + 3 y = 2x 2 5x + 3 y = 2x 2 + 5x + 3 y = 2x 2 + 2x + 3 y = 2x 2 + 5x + 3 y = 2x 2 + 3

5 Chapter 6: Quadratic Functions Section 6.1 Summary of Quadratic Function in Standard Form The degree of all quadratic functions is 2. The standard form of a quadratic function is y = ax 2 + bx + c, where a 0. The graph of any quadratic function is a parabola with a single vertical line of symmetry. A quadratic function is written in standard form y = ax 2 + bx + c, has the following characteristics: The highest or lowest point on the graph of the quadratic function lies on its vertical line of symmetry. If a is positive, the parabola opens up. If a is negative, the parabola opens down. a > 0 a < 0 Changing the value of b changes the location of the parabola s line of symmetry. The constant term, c, is the value of the parabola s y-intercept.

6 Chapter 6: Quadratic Functions Section 6.2 Section 6.2: Properties of Graphs of Quadratic Functions Terminology: Vertex: The point at which the quadratic function reaches its maximum or minimum value. Axis of Symmetry: A line that separates a two dimensional figure into two identical parts. Ex. A parabola has a vertical axis of symmetry passing through the vertex. Maximum Value: The greatest value of the dependent variable in a relation. Minimum Value: The least value of the dependent variable in a relation. Domain: All possible x-values that a graph passes through. Range: All possible y-values that a graph passes through. Using Symmetry to Estimate the Coordinates of the Vertex Ex. Nicole plays on her school s volleyball team. At a recent match, her friend, Marko, took some time-lapse photographs while she warmed up. He set his camera to take pictures every 0.25 seconds. He started his camera at the moment the ball left her arms during a bump and stopped the camera at the moment that the ball hit the floor. Marko wanted to capture a photo of the ball at its greatest height. However, after looking at the photographs, he could not be sure that he had done so. He decided to place the information from his photographs in a table of values. Time (s) Height (ft) From his photographs, Marko observed that Nicole struck the ball at a height of 2 ft above the ground. He also observed that it took about 1.25 seconds for the ball to reach the same height on the way down. When did the volleyball reach its greatest height?

7 Chapter 6: Quadratic Functions Section 6.2 Ex. Given the coordinates of a parabola, as shown in the table, determine the coordinates of the vertex. x y Determining the Coordinates of the Vertex from Standard Form There exists a relationship between the value of the x-coordinate of the vertex and the values of coefficients in standard form. To determine the value of the x component of the vertex (p) we use the formula: x = b 2a To determine the y component of the vertex (q) we simply plug in the result of the above formula into the given equation in standard form. Ex: For each of the following state the vertex, directionality, y-intercept, and range: (a) f(x) = x 2 + 2x + 8 (d) f(x) = 2x 2 12x + 25 (b) f(x) = x 2 2x (e) f(x) = 3x 2 + 9x + 12

8 Chapter 6: Quadratic Functions Section 6.2 (c) (x) = 4x 2 2x + 5 (f) f(x) = x x 4 Determining the Maximum Value of a Quadratic Ex. Some children are playing at the local splash pad. The water jets spray water from ground level. The path of water from one of these jets forms an arch that can be defined by the function: f(x) = 0.12x 2 + 3x Where x represents the horizontal distance from the opening in the ground in feet and f(x) is the height of the sprayed water, also measured in feet. What is the maximum height of the arch of water, and how far from the opening in the ground can the water reach? State the range for the jet of water. Ex. A skier s jump was recorded in frame-by-frame analysis and placed in one picture. The skier s coach used the picture to determine the quadratic function that relates the skier s height above the ground, y, measured in metres, to the time, x, in seconds that the skier was in the air: y = 4.9x x + 1 Determine the skier s maximum height, to the nearest tenth of a metre, and state the range of the function for this context.

9 Chapter 6: Quadratic Functions Section 6.2 Graphing a Quadratic Function in Standard Form Ex: For each of the functions determine the following functions then Sketch: 1. The Vertex 2. The Domain and Range 3. The Direction of Opening 4. The Equation of the Axis of Symmetry (a) y = 2x 2 + 4x 1 (a) y = 2x 2 + 4x 1

10 Chapter 6: Quadratic Functions Section 6.2 (b) y = 1 4 x2 + 2x 3 (c) y = 1 2 x2 2x 2 15 y x

11 Chapter 6: Quadratic Functions Section 6.2 (d) y = 3x 2 6x 15 y x

12 Chapter 6: Quadratic Functions Section 6.3 Section 6.3: Factored Form of a Quadratic Function Terminology: Factored Form of a Quadratic Function: A quadratic function is in factored form when it is written in the form: y = a(x r)(x s) Where a, r, and s are real numbers and a 0. Zero: In a function, a value of the variable that makes the value of the function equal to zero. Zeros are also referred to as solutions, roots, and x-intercepts. The zeros of an equation can be determined by setting an equation equal to zero (or by substituting y = 0 into an equation). Y-Intercept: The value at which a function passes through the y-axis. The x-coordinate of y-intercept is always zero and hence can be found from an equation by plugging x = 0 into an equation. Determining the Zeros of an Equation in Factored Form Ex. Determine the zeros of each equation (a) y = 3(x + 2)(x 5) (b) y = 1 (x + 3)(x + 9) 2 (c) f(x) = 1 (x)(x 4) (d) f(x) = 5(x 1)(x 1) 7

13 Chapter 6: Quadratic Functions Section 6.3 (e) y = (x + 10)(x 10) (f) y = 2(3x + 4)(x 5) Determining the Zeros of Quadratic Functions in Standard Form Determine the zeros of a function by using partial factoring (a) y = x 2 + 8x + 15 (b) y = 3x 2 + 6x 3 (c) y = x 2 6x 40 (d) y = 2x x + 12 Graphing a Function in Factored Form Graph each of the functions. (a) y = 2(x + 1)(x 2)

14 Chapter 6: Quadratic Functions Section 6.3 (b) y = 1 (x 8)(x 3) 4 (c) y = 1 (x + 6)(x 2) 2 15 y x

15 Chapter 6: Quadratic Functions Section 6.3 (d) y = 3(x + 1)(x 2) 15 y x (e) y = 1 (x + 5)(x 7) 5

16 Chapter 6: Quadratic Functions Section y x Determining the Equation of Quadratic Function in Factored Form from a Graph To determine the equation of a quadratic function in factored form y = a(x r)(x s) we need to determine the x-intercepts and y-intercept. The x-intercepts fill in the r and s values and then use the y-intercept to solve for a.

17 Chapter 6: Quadratic Functions Section 6.3 (a) (b)

18 Chapter 6: Quadratic Functions Section 6.3 (c) (d)

19 Chapter 6: Quadratic Functions Section 6.3 Max/Min Word Problems using Factored Form Quadratics Ex. The members of a Ukrainian church hold a fundraiser every Friday night in the summer. They usually charge $6 for a plate of perogies. They known from previous Fridays, that 120 plates of perogies can be sold at $6 but, for every $1 the price is increased, 10 fewer plates will be sold. What should the members charge if they want to raise as much money as they can? Ex. A career and technology class at a high school operates a small T-shirt business out of the school. Over the last few years, the shop has had monthly sales of 300 T-shirts at a price of $15 per T-shirt. The students have learned that for every $2 increase in price, they will sell 20 fewer T-shirts each month. What should they charge for their T-shirt to maximize their monthly revenue? What will their monthly revenue be? Ex. Paulette owns a store that sells used video games in Red Deer. She charges $10 for each used game. At this price she sells 70 games per week. She has learned that for every $1 increase in price, she sells 5 fewer games per week. At what price should Paulette sell her games to maximize her sales? What is her maximum revenue?

20 Chapter 6: Quadratic Functions Section 6.3 Ex. You wish to build a rectangular garden next to your house. You intend to use your house as one of the sides of the garden. If you buy 30 ft of fencing to go around the edges of your garden, determine the length and width that will result in your garden having a maximum area. Ex. Nathan is planning on building a sand box next to his house to play in. He plans to use his house as one side of the sandbox. He has purchased 20 ft of board to create the sides of his sandbox. Determine the length and width of the sandbox to produce a maximum area. Ex. Mr. Hutchings is planning on creating a kennel for his classroom to store some of his troublesome Math 2201 students when they need a time out. The design for the kennel is shown below. If he has 34 ft of boards to create the kennel, what are the measurements of each kennel room that will produce the maximum area? Kennel

21 Chapter 6: Quadratic Functions Section 6.4 Section 6.4: Vertex Form of a Quadratic Function Terminology: Vertex Form of a Quadratic: A quadratic function is in factored form when it is written in the form: y = a(x h) 2 + k Where a, r, and s are real numbers and a 0. In vertex form, the values of h and k are the coordinates of the vertex of the function (h, k). Examples of vertex form are shown below, state the vertex of each: (a) y = (x + 1) 2 2 Vertex: (b) y = (x 6) Vertex: (c) y = 2(x + 4) 2 4 Vertex: (d) y = 1 (x + 10)2 Vertex: 2 (e) y = 4(x 3) (f) y = 3x (g) y = 1 9 (x 1.5)2 4 (h) y = x 2 Vertex: Vertex: Vertex: Vertex: Sketching a Graph of a Quadratic from Vertex Form To sketch a graph of a quadratic from vertex form, we need two pieces of information: 1. The coordinates of the vertex 2. Any other set of coordinates relating to the graph of the function, usually something simple like the y-intercept is recommended! We then place those coordinates on the graph, create a third point using the axis of symmetry as we did in the previous sections, and then sketch the graph of the function passing through those three points. Ex. Sketch the graph of each function in vertex form. (a) y = 2(x + 3) 2 10

22 Chapter 6: Quadratic Functions Section 6.4 (b) y = 1 2 (x 4)2 + 5 (c) y = 5(x 1) 2 5

23 Chapter 6: Quadratic Functions Section 6.4 (d) y = 6x 2 4 (e) y = 1 (x + 5)2 10

24 Chapter 6: Quadratic Functions Section 6.4 Determining the Equation of a Quadratic in Vertex Form When determining the equation of a quadratic in vertex form, we need the location of the vertex and one other point. The additional point is usually one that is easily read or one that is clearly identified. We plug the vertex (h, k) into the equation y = a(x h) 2 + k, then use the additional point (x, y) into the equation and solve for a. Ex. Determine the equation in vertex form for each graph (a) (b)

25 Chapter 6: Quadratic Functions Section 6.4 (c) (d)

26 Chapter 6: Quadratic Functions Section 6.4 Using Vertex and Directionality to Determine the Number of Zeros To determine the number of zeros that a quadratic function has, we use the vertex and the directionality of a given function. Drawing a sketch often helps to determine the number of zeros. Once you have a sketch, you just count the number of x-intercepts that your sketch has. Note that there are only three possibilities for a quadratic function; Two Zeros, One Zero, or No Zeros. Ex. Determine the number of zeros for each function: (a) y = (x + 5) 2 6 (b) y = (x 6) (c) y = 2(x + 4) 2 (d) y = 1 2 (x + 10)2 6 (e) y = 4(x 3) (f) y = 3(x + 1) 2 (g) y = 1 9 (x 1.5)2 4 (h) y = x 2

27 Chapter 6: Quadratic Functions Section 6.4 Word Problems and Vertex Form To solve word problems involving vertex form we often have to create the vertex form equation y = a(x h) 2 + k and then use the equation to answer the question. As usual we will need two coordinates to solve the equation, one of which needs to be the vertex of the equation. Ex. A soccer ball is kicked from the ground. After 2 s, the ball reaches its maximum height of 20 m. It lands on the ground at 4 seconds. (a) Determine the equation of the quadratic function in vertex form that models the height of the ball after it is kicked. (b) What was the height of the ball after 1 second? (c) When was the ball at the same height on the way down? Ex, When playing a game of baseball, a player hits after it is pitched. It reached a maximum height of 24.2 m after 6.5 seconds. The ball was in there air for 13.8 seconds. (a) Determine the equation of the quadratic function in vertex form that models the height of the ball after it is hit. (b) Determine the height of the ball after 9 seconds. (c) When was the ball at the same height on the way up?

28 Chapter 6: Quadratic Functions Section 6.4 Ex. The underside of a concrete underpass forms a parabolic arch. The arch is 30.0 m wide at the base and 10.8 m high in the centre. What would be the headroom at the edge of a sidewalk that starts 1.8 m from the base of the underpass? Would this amount of headroom be safe for walking? Ex. The Lions Gate Bridge in Vancouver is a suspension bridge. The main span, between the two towers, is 472 m long. Large cables are attached to the top of both towers, 50 m above the road. Each large cable forms a parabola. The road is suspended from the large cables by a series of vertical cables. The shortest vertical cable measures about 2 m from the road. Use this information to determine the quadratic function in vertex form to model this situation. Use your equation to determine the height of the vertical cable that is 300 m away from the first tower.

29 Chapter 6: Quadratic Functions Section 6.4 Ex A soccer ball is kicked into the air and first lands on the ground 20 m away. If a maximum height of 8 m is reached, determine the vertex form of the quadratic function representing the path of the soccer ball while it is in the air. y Path of Soccer Ball (0,0) (20,0) x Ex. The entrance to a garden party is in the shape of an arch. If the arch is to be 2 metres wide and 2 metres tall write the quadratic equation that describes its shape. Write the equation that models this situation in vertex form.