Solve each equation. To analyze and manipulate quadratic models to identify key information about a relationship or real world situation.

Transcription

1 Test Yourself Solve each equation. Lesson 13 Problem Solving with Quadratic Functions Goals To analyze and manipulate quadratic models to identify key information about a relationship or real world situation. To create quadratic models to represent real life relationships and situations.

2 Analzying Quadratic Models In these types of problems you are given the function that represents the scenario. You will be asked to find a specific piece of information. G: What form of the function is given? What is the dependent variable ie. f(x)? What is the independent variable (x)? Can I sketch the graph? R: What are they asking me to find? Is this a key feature? (ie. root(s), vertex, y intercept)? Am I solving for y given x? Am I solving for x given y? A: Factored Form Vertex Form Standard Form

3 S: Show all your steps. Explain your thinking where needed. P: State your final answer. Reflect on its "reasonability". Use estimation and your understanding of the model to determine if your answer "makes sense". Example 1 A group of churchgoers are planning their annual pie fundraiser. A retired mathematician has analyzed their finances and determined that the profit per day (in dollars), based on the number of pies sold, x, is represented by the function P(x) = 2x x 600. a. What is the profit if they sell 15 pies?

4 b. How many pies do they need to sell in order to break even? c. How many pies do they need to sell in order to profit $100? d. What does the y intercept for this function represent? e. How many pies do they need to sell to make the maximum profit each day, and how much will they earn?

5 Problem 1 You try! Refer to Example 1 A base jumper jumps from a trampoline off a cliff and parachutes to the ground. His height, h(x), with respect to the horizontal distance he travels from the base of the cliff, x, is described by the function. a. What is the height of the jumper when he is on the cliff? b. At what distance from the base of the cliff does the base jumper land? c. What is the maximum height the base jumper reaches? Creating Quadratic Models In certain types of problems we will be required to create algebraic expressions to represent a given relationship, that we can then represent as an equation. In doing so, a quadratic function is created in some instances. We then need to use our understanding of the quadratic function to manipulate and/or identify key pieces of information.

6 Example 2 If we have 40 m of stone edging to place around the perimeter of a rectangular flower bed, what is the maximum area of the flower bed that we can create? Example 3 A concert promoter knows that if they sell tickets for $30 per ticket they will sell 500 tickets. He also knows from experience that for every $1 increase in ticket price, 10 fewer tickets will be sold. a. Create a function to model the revenue based on the ticket. b. What ticket price will maximize the profit?

7 Creating Quadratic Models In these types of questions we will be given information that we can use to create, or construct, a quadratic function. We can create a model using factored form or vertex form, depending on the information we have. Factored Form Vertex Form Example 4 Marie kicks a soccer ball and it reaches a maximum height of 5 m after traveling a horizontal distance of 18 m. Determine the equation that represents the height of the ball, h(x), with respect to the distance it travels down the field, x. Then determine the distance at which the ball lands.

8 Steps for Creating a Quadratic Model Step 1 Identify the variables you want to compare and which one is dependent (y) and which one is independent (x). Step 2 Draw a sketch of the scenario. Step 3 Identify which form of the quadratic function you should use, based on the information you know. Roots Factored Form Vertex Vertex Form Step 4 Fill in the information you know. Step 5 Solve for a using another known point. Step 6 Write your final equation and check with graphing software to make sure it is accurate. Let's look at some more examples. p. 189

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10 Consolidate Try each question at your table. Compare your answers when complete. Problem 2 Refer to Example 2 Marjorie has 200 m of fencing to enclose her pet llama. What dimensions should she choose to create her pen so that her llama has the most space inside?

11 Problem 3 Refer to Example 4 A baseball player hits a home run and the ball reaches a maximum height of 17 m when it is approximately 70 m away from the hitter. The player made contact with the ball at a height of 1.2 m from the ground. Create the model that represents the height of the baseball for a given horizontal distance. Then determine the distance the ball traveled. Practice Section 3.2, page 154, #8, 9, 11, 15 Section 3.5, page 178, #7 10, 13, 14 Section 3.7, page , #10, 11 Looking for Extensions? Page 178, #12, 16, 17 Page 193, #16