Quadratic Functions. Chapter Properties of Quadratic Functions... p Investigating Quadratic Functions... p. 6 in Vertex Form: Part 1

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1 Chapter 3 Quadratic Functions 3. Properties of Quadratic Functions p Investigating Quadratic Functions p. 6 in Vertex Form: Part Investigating Quadratic Functions p. 11 in Vertex Form: Part 3.3 Completing the Square: Part p Completing the Square: Part p Graphical solutions of Quadratic p. 6 Equations Ch. 3 Review p. 3 Loo/Stewart/Lee/Ko Pre-Calculus 11

2 3. Properties of Quadratic Functions Recall: For example: Quadratic Function A quadratic function is any function that can be written in the form and c are constants and, where a, b A quadratic function can be written in different forms: Standard Form: Vertex Form: Factored form: Example 1: function. quadratic Use a table of values: x y - Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 1 of 37

3 Identify Quadratic Functions: Quadratic Functions? yes/no The Graph of Quadratic Functions: The graph of every quadratic function is a curve called a. The of a parabola is its highest or lowest point. The vertex may be a point or a point. The intersects the parabola at the vertex and its equation is equal to the -coordinate of the vertex. The parabola is symmetrical about this line. Example : Identify the following: a) y x Equation of an Axis of Symmetry Coordinates of the Vertex: x- intercept: y-intercept Domain: Range: Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page of 37

4 b) Equation of an Axis of Symmetry Coordinates of the Vertex: x- intercept: y-intercept: Domain: Range: c) 1 Equation of an Axis of Symmetry 3 Coordinates of the Vertex: x- intercept: y-intercept: Domain: Range: Example 3: Use technology to graph the function. Find the following: a) Vertex b) Axis of symmetry c) Direction of opening d) Maximum or Minimum values e) Domain f) Range x [, ] y [, ] g) Any intercept(s) Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 3 of 37

5 Example 4: A stone is dropped from a bridge into a river. The height of the stone, h metres, above the river, t seconds, after it is dropped is modeled by the equation a) Graph the function. b) Did the stone hit the river? c) What is the domain? What does it represent? x [, ] y [, ] Assignment Page 174, #1 3, 5ab (gc), 6ab, 7, 8, 10, 1 (gc) (gc) requires a graphing calculator Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 4 of 37

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7 3.1 Investigating Quadratic Functions in Vertex Form: Part 1. Use a table of values: x y Also do it on your graphing calculator. Now graph on the same grid. What do you notice about the -intercept? Now graph on the same grid. What do you notice about the -intercept? Therefore, comparing with, the graph moves. If If > 0, the graph moves. < 0, the graph moves. Graph y x. What do you notice about the x-intercept when compared to? Graph. What do you notice about the x-intercept when compared to? Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 6 of 37

8 Therefore, comparing y x p with, the graph moves. If If > 0, the graph moves. < 0, the graph moves. Example 1: Graph the function y x 1 3. Determine the following questions below: a) How is the graph transformed from b) What is the vertex? c) Identify the equation of the axis of symmetry d) Does it have a minimum or maximum value? What is the value? e) Domain f) Range For the graph y x p q, the coordinates of the vertex are. Example : Write an equation of a quadratic function with vertex 5, 8. Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 7 of 37

9 Consider the following graphs. 1) ) 3) 4) 5) a) Graph the following functions: 1,, 4 What do you notice about the direction of opening? b) Graph the following functions: 3, 5 What do you notice about the direction of opening? c) Graph the following functions: 1 and Is the graph vertically expanded (thin) or vertically compressed (wide)? d) Graph the following functions: 1 and 4 Is the graph vertically expanded (thin) or vertically compressed (wide)? Comparing with a a. If or, graph is thin ( ). If or, graph is wide ( ). Example 3: Write the equation of the parabola that opens down and is congruent to y 1 x 3 Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 8 of 37

10 The Vertex form of a quadratic function Given the function: a (p, q) are the coordinates of the vertex is the equation of the axis of symmetry Example 4: Graph the following equations on the same grid. Assignment Page 157, #, 3ab, 4ad Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 9 of 37

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12 3.1 Investigating Quadratic Functions in Vertex Form: Part The Vertex form of a quadratic function Given the function: a (p, q) are the coordinates of the vertex is the equation of the axis of symmetry Example 1: Write an equation for a parabola with vertex and -intercept of 5. Example : For the equation a) vertex b) direction of opening y x 3, sketch the graph and find the following: c) domain d) range e) max/min value f) axis of symmetry g) y-intercept h) x-intercept Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 11 of 37

13 Example 3: For the equation, sketch the graph and find the following: a) vertex b) direction of opening c) domain d) range e) max/min value f) axis of symmetry g) y-intercept h) x-intercept Example 4: A company makes T-shirts. The profit, P dollars, for selling a certain style of T-shirt is given by the equation P 0( x 5) 5780, where x dollars is the selling price of one T-shirt. a) Determine the coordinates of the vertex of the function. b) What is the maximum profit possible for this company? c) What price do the T-shirts sell at to get the maximum profit? Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 1 of 37

14 Example 5: During a game of tennis, Serena hits the tennis ball into the air along a parabolic trajectory. Her initial point of contact with the tennis ball is 1 m above the ground. The ball reaches a maximum height of 10 m before falling toward the ground. The ball is again 1 m above the ground when it is m away from where she hit it. Write a quadratic function to represent the trajectory of the tennis ball if the origin is on the ground directly below the spot from which the ball was hit. Assignment Page 157, #5, 7ac, 8ab, 9bc, 13, 18, 1a Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 13 of 37

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16 3.3 Completing the Square: Part 1 Recall: The vertex form of the quadratic equation: y a x p q Sometimes quadratic functions are not written in vertex form. They are written in the standard form. So to convert from standard form to vertex form, we apply a method called. Exercise 1: What number must you add to make the following a perfect square? a) x 6x b) x 10x Exercise : What number must you add to make the following a perfect square? We will use algebra tiles to Determine the value that is needed to be added to complete the square. x 3x Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 15 of 37

17 Example 1: Rewrite the function (standard form) to vertex form. Completing the Square: Modeling with Algebra Tiles Algebra Tiles Method Model the polynomial, using algebra tiles. Algebraic Method Are you able to form a square using these tiles? If not, what do you need to do? Idea of Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 16 of 37

18 Example : a) Re-write into vertex form by completing the square. Extension: b) What is the vertex? c) Is it a maximum or minimum? d) What is the max/min value? e) the max/min occur? term) does not equal 1? Example 3: Re-write the following functions from standard from to vertex form. a) b) Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 17 of 37

19 Example 4: Convert the following into vertex form. a) y 3x 1x 35 b) 1 Example 5: Graph converting it to vertex form. y x x 3 by first Find the maximum or minimum value. State the domain and range. Assignment Page 19, #-5 (ac), 6bc, 7ce, 8ac, 1cd Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 18 of 37

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21 3.3 Completing the Square: Part This section involves word problems that translate into quadratic functions. You will be required to find the max/min value of a function by manipulating the quadratic into its vertex form. Do you remember which variable will give you the maximum/minimum value???? Answer: Recipe for success: 1) Determine the item that is to be maximized or minimized. ) Define variables for all other quantities 3) Write an equation involving your variable(s) and the item that is to be maximized/minimized. 4) If there are two variables, eliminate one of the variables using substitution. 5) Complete the square to write the equation in vertex form y a x p q 6) The max/min value of the function is the -coordinate of the vertex. Example 1: meters, above the water is given by, where is the time in seconds after the diver leaves the board. a) What is the maximum height of the diver in metres? b) When does he reach that height? Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 0 of 37

22 Example : Find two numbers such that their difference is 4 and their product is a minimum. Example 3: You are creating a rectangular enclosure using 180 metres of fencing. However, one side of the enclosure is a barn wall. Find the dimensions of the enclosure that will maximize the area. Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 1 of 37

23 Example 4: A rectangular field is to be enclosed by a fence and then divided into three smaller plots by two fences parallel to one side of the field. If there are 900 metres of fence to use, find the dimensions and the maximum area of the field. Example 5: 300 people will attend a concert when the admission price is $0. The attendance decreases by 10 people for each $1 increase in the price. a) What price of admission will yield the maximum revenue b) What is the maximum revenue? Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page of 37

24 Assignment Page 194, #15, 18, 19,, 3, (worksheet), 4 6 Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 3 of 37

25 3.3 Part Homework worksheet 4. The sum of two natural numbers is 16. What are the numbers if their product is a maximum? 5. The sum of two numbers is 0. a) Find the numbers if the sum of their squares is a minimum. b) What is the minimum product? 6. The sum of a number and three times another number is 4. Find the numbers if their product is a maximum. Ans: 4. 6 and 6 5a) 10 and 10 b) and 1 Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 4 of 37

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27 4.1 Graphical Solutions of Quadratic Equations Recall: A quadratic function is a second degree polynomial function that can be written in the form: where,b and are constants, and. We have worked with two forms of the quadratic function so far: We will now look our final form of a quadratic function and see how it relates to solving the corresponding quadratic equation. Let us now compare a quadratic function to its corresponding quadratic equation: Example 1: a) Solve the following quadratic equation b) Determine the x-intercepts of the following quadratic function. Root(s) of an equation:. Zero(s) of a function:. x-intercept(s) of a graph:. Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 6 of 37

28 Example : Quadratic Equations with Two equal Real Roots: Given = x 4x 4 x- intercept(s) of graph: zero(s) of the function, root(s) of the equation, value of the discriminant: Quadratic Equations with Two Different Real Roots: Given x- intercept(s) of graph: zero(s) of the function, root(s) of the equation, Value of the discriminant: Quadratic Equations with No Real Roots: Given x- intercept(s) of graph: zero(s) of the function, root(s) of the equation, x x 4 0 Value of the discriminant: Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 7 of 37

29 Example 3: Solve the equation x x 4 7 by graphing. Method One: Graph: Graph: Find the x-coordinates of the points of intersection. Method Two: Rearrange the equation x x 4 7 to Graph: Find the x-intercepts of the graph Example 4: Determine the zeroes of the function algebraically. Round to the nearest tenth if applicable a) b) Example 5: The manager of Clothing Boutique is investigating the effect that raising or lowering dress prices has on the daily revenue from dress sales. The function revenue R, in dollars, from dress sales, where is the price change, in dollars. What price changes will result in no revenue? R x Price Change ($) Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 8 of 37

30 There are certain quadratic functions that can be written in factored form:. Using the fact that a quadratic function is symmetrical we can use the zeros of the function to help determine the maximum/minimum value and where it occurs. To determine the equation of the axis of symmetry we can calculate the average value of the zeros. Equation of axis = of symmetry Recall that the value of the axis of symmetry is equal to the x-coordinate of the vertex. So after you find this value (x), you can substitute it into your function to find the maximum/minimum value (y). Example 6: The sum of two numbers is 40. Their product is a maximum. a) Determine the numbers that produce the maximum product. b) What is the maximum product? Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 9 of 37

31 Example 7: 0 metres of fencing is available to enclose a rectangular area. a) What is the maximum area that can be enclosed? b) What dimensions produce the maximum area? c) State the domain and range for this problem. Assignment Page 15, #1, 3ade, 5 7, 10, 13 Page 195, #18, 3a (Solve by using zeros, NOT by completing the square) Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 30 of 37

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33 Ch. 3 Review Properties of Quadratic Functions o Graphing Quadratic Functions Completing the Square Max/Min problems Word problems Solving quadratic equations graphically Example 1: Given the following quadratic function, answer the following : a) Coordinates of the vertex : b) direction of opening : c) Domain : d) Range : e) Equation of axis of : f) Maximum or Minimum : symmetry value g) y-int : h) x-intercept : Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 3 of 37

34 Example : Graph the following functions on the same grid. a) b) Example 3: Determine the range of the Example 4: Determine the coordinates of function. the vertex of: Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 33 of 37

35 Example 5: Given the function a) Determine the zeros of the function. b) Determine the max/min value without completing the square. Example 6: Two numbers have a difference of 6 and the sum of their squares is a minimum. Determine the numbers. Example 7: A bridge spans a horizontal distance of 40 m and has a parabolic arch above it. One metre from the edge of the bridge, the arch is 1.95 metres high. a) Determine an equation that represents this parabolic arch. b) How high is the arch at the centre of the bridge? Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 34 of 37

36 Example 8: A candy company collected data on sales from the previous year. When their candy bar was priced at $1.80, they sold 1600 of them. Market research determined that if they lowered the price of the candy bar by $0.05, they would increase sales by 80 bars. a) What price will maximize revenue? b) What was the maximum revenue? Assignment Page 198, #1cd,, 3bc, 4ac, 5 7, 9a, 1, 14bc, 15, 17 Page 58, #1cd, 3, 5 Ch. 3 Quadratic Functions Loo/Stewart/Lee/Ko Page 35 of 37

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