PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 1 Graphs of Quadratic Functions (.1) Date A. QUADRATIC FUNCTIONS A quadratic function is an equation that can be written in the following form: ax bx c OR f ( x) ax bx c where a, b, and c are constants and a 0 When ou graph a quadratic function, ou get a curve, called a. B. PRACTICE Which of the following are quadratic functions? Hint: You ma need to rearrange the equation so it looks like a quadratic function.) 1. x 6 4. 1 x 4. g( x) x 5. h ( t) ( t 1) 5 3. f ( x) (x 5)( x 6) C. VOCABULARY Ever parabola can be divided into two smmetrical halves. This line is called the and is designated b a dotted line in the graph. The point where the line of smmetr intersects the parabola is called the. The - is where the parabola intersects the -axis and the - are the two points where the parabola intersects the x-axis. Chapter Quadratic Functions 1
Recall that the is the set of x-values represented b the graph or the equation of a function. The is the set of -values for which the graph or equation is true. Example 1: Graph the function defined b the equation x 7x 10. Determine the -intercept, the x- intercepts, the equation of the axis of smmetr, the coordinates of the vertex, and the domain and range. x -1 0 1 3 4 5 6 7 8 -intercept: x-intercepts: Chapter Quadratic Functions
axis of smmetr: vertex: domain: range: D. QUADRATIC EQUATIONS A quadratic equation is an equation that can be written in the following form: ax bx c 0 where a, b, and c are constants and a 0 The x-intercepts of a quadratic equation are called the of the equation (whereas the x- intercepts of a quadratic function are called the ). Example : Using algebra, determine the roots of the quadratic equation x 7x 10 0. Example 3: Solve the quadratic equation 3x 11x 4 0. Chapter Quadratic Functions 3
1 Example 4: Write the equation of a quadratic function that has zeroes and. 3 Example 5: The zeroes of a quadratic function are 4 3 and 5. Write the equation for this function. Chapter Quadratic Functions 4
PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson Modelling Real Situations Using Quadratic Functions (.) Date The use of quadratic functions in real life can be seen when studing projectile motion in phsics. We also see quadratic functions in situations where a certain quantit is the product of two other quantities. For example, the area of a rectangle is the product of its length and its width. When one quantit increases, the other decreases. Example 1: You have 40 m of fencing to make a rectangular pen for our dogs. a) Represent the area of the pen as a function of the length of one side of the pen. b) Graph the function x 0 5 8 10 1 15 0 c) What dimensions provide an area greater than 90 m? Chapter Quadratic Functions 5
Example : A compan makes canoes, and then sells them for $500 each. At this price, it can sell 60 canoes in a season, generating revenue of $30 000. To increase revenue, management is planning to increase the selling price. It estimates that for ever $50 increase in price, the number of canoes sold will drop b 4. a) Represent the number of canoes sold as a function of the selling price. b) Represent the revenue as a function of the selling price. c) Sketch the function without making a table of values. (Hint: Determine the roots and the coordinates of the vertex.) d) What selling price will provide the maximum revenue? What is the maximum revenue? e) What range of selling prices will provide revenue greater than $30 000? Chapter Quadratic Functions 6
PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 3 Graphing a( x p) q (.3) Date The simplest quadratic function is Example 1: Graph x. x. x The graph parabola is: x can be expanded or compressed or moved horizontall or verticall. The general equation for a a( x p) q Chapter Quadratic Functions 7
Example : Graph the following functions and label them. x x x x 1 x 1 x Example 3: Graph the following functions and label them. x ( x 4) ( x 4) Example 4: Graph the following functions and label them. x x x Chapter Quadratic Functions 8
Example 5: Graph the function ( x 3). Does the parabola open up or down? x Is the parabola expanded or compressed? The coordinates of the vertex is The equation of the axis of smmetr is Example 6: Graph the function f ( x) ( x 4) 3. Does the parabola open up or down? x Is the parabola expanded or compressed? The coordinates of the vertex is The equation of the axis of smmetr is Example 7: Determine the equation of the following graph of a quadratic function. Chapter Quadratic Functions 9
PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 4 Graphing ax bx c (.4) Date A. COMPLETING THE SQUARE Recall: The general equation for a parabola is: a( x p) q Also, the equation for a parabola can be written in the form: ax bx c It is easier to graph the first equation rather than the second one. However, sometimes it is necessar to graph functions that are written in the second form. To do this, we need to complete the square. Example 1: Write the following functions in the form a) x 8x 9 a( x p) q. Step 1: If necessar, remove the coefficient of x as a common factor from the first two terms. Step : Take the coefficient of x (including the negative sign, if present), divide b and then square it. Add and subtract this number inside the brackets. Step 3: Remove the last term from the brackets and combine with the constant term. Step 4: Factor the expression in the brackets as a complete square. b) ( x) 3x 1x 8 f c) f ( x) 5x 30x 7 Chapter Quadratic Functions 10
Example : Write x 1x 11 in the form a x p) q (, then sketch the graph. Complete the square: x B. MAXIMUM AND MINIMUM VALUES The vertex represents the maximum or minimum value of a function. You can determine this value without drawing a graph. Simpl look at the equation in the form a( x p) q. Example 3: Compare the following equations and determine their maximum or minimum values. a) 3( x 4) 7 What are the coordinates of the vertex? Is a positive or negative? Does the graph open upward or downward? Is the vertex the maximum or minimum value of the graph? b) f ( x) ( x 3) 8 What are the coordinates of the vertex? Is a positive or negative? Does the graph open upward or downward? Is the vertex the maximum or minimum value of the graph? Chapter Quadratic Functions 11
PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 5 Maximum and Minimum Problems (.5) Date In man problems involving the maximum or minimum value of a function, the function is not given. It must be determined using the information that is given in the problem. Example 1: Two numbers have a difference of 10. Their product is a minimum. What are the numbers? Step 1: Write our let statements. Step : Write an algebraic expression. Step 3: The algebraic expression must contain onl one variable. If it contains more that one variable, substitute equivalent expressions into the equation. Step 4: Determine whether the quadratic function has a maximum or minimum value. Then complete the square to determine this value and where it occurs. Step 5: Answer the question in the problem. Chapter Quadratic Functions 1
Example : A rectangular piece of land is bounded on one side b a river and on the other three sides b a total of 80 m of fencing. Determine the dimensions of the largest possible piece of land. Chapter Quadratic Functions 13
Example 3: Computer software programs are sold to students for $0 each. Three hundred students are willing to bu them at that price. For ever $5 increase in price, there are 30 fewer students willing to bu the software. What is the maximum revenue? Chapter Quadratic Functions 14
PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 6 The Inverse of a Linear Function (.6) Date The inverse of a function is a relation whose rule is obtained from that of a function b interchanging x and. To determine the inverse of a function: Interchange x and in the equation of the function. Solve the resulting equation for. Example 1: Determine the equation of the inverse of the linear function 3x. When x and are interchanged in the equation of a function: The x and coordinates of the points that satisf the equation of the function are interchanged. The graph of the function is reflected in the line x. Example : Determine the equation of the inverse of the linear function x 4. Then make a table of values for both linear functions and graph then on the same graph. x 4 : Inverse of x 4 : x x 0-4 0 4 4 Chapter Quadratic Functions 15
To express the inverse of a linear function f (x) in function notation, we use the smbol 1 ( x ) f (x) f of x f 1 ( x) f inverse of x f. Example 3: Determine the inverse of the function f ( x) x 5. 1 Example 4: Given the graph of f (x), graph f 1 ( x ) on the same grid. Chapter Quadratic Functions 16
PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 7 The Inverse of a Quadratic Function (.7) Date Last lesson, we looked at the inverse of a linear function. To determine the inverse of a quadratic function, we use the same steps. Note: We do not use the notation f 1 ( x ) necessaril be a function! to represent the inverse of a quadratic function because it ma not Example 1: Consider the function f ( x) x 4. a) Determine the inverse of f (x). Graph f (x) and its inverse on the same grid. f ( x) x 4 : x b) Is the inverse of f (x) a function? Explain. Chapter Quadratic Functions 17
Sometimes it is convenient to restrict the domain of a quadratic function so that its inverse is a function. Example : Show two was to restrict the domain of f ( x) x 4 so that its inverse is a function. Illustrate each wa with a graph. Chapter Quadratic Functions 18
PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 8 Solving Linear-Quadratic Sstems Algebraicall Date To solve a linear-quadratic sstem algebraicall, we need to follow these steps: Step 1: Solve the linear equation for one variable. (Usuall, we solve for.) Step : Substitute into the quadratic equation and solve for the other variable. Step 3: Substitute the results from Step into the linear equation and solve for the first variable. Example 1: Find the coordinates of the points of intersection of the circle 10 Check the solution. x and the line 3 6 x. Chapter Quadratic Functions 19
Example : Solve the following linear-quadratic sstem: x x 6 Example 3: Solve the following linear-quadratic sstem: x 1 x 4 Chapter Quadratic Functions 0
Example 4: Solve the following linear-quadratic sstem: x x 1 Example 5: Solve the following linear-quadratic sstem: x x 3 Chapter Quadratic Functions 1