Unit 6 Part 2 Quadratic Functions 2/28/2017 3/22/2017

Transcription

1 Unit 6 Part 2 Quadratic Functions 2/28/2017 3/22/2017 Name: By the end of this unit, you will be able to Simplify radical expressions using properties of square roots Solve quadratic equations using the Quadratic Formula Use the discriminant to determine the number of solutions to a quadratic equation Identify linear, quadratic, and exponential functions from given data Write equations that model data

2 Table of Contents Simplifying Radical Expressions... 3 Product Property of Square Roots... 3 Simplifying Square Roots... 3 Multiplying Square Roots... 4 Simplifying Square Roots with Variables... 4 Quotient Property of Square Roots... 5 Rationalizing Denominators Completing the Square... 7 The Projectile Motion Model Quadratic Formula The Discriminant Successive Differences... 15

3 6.2.1 Simplifying Radical Expressions Warm Up: 1. List all the factors of List all the factors of What is the biggest square number that is a factor of 80? 4. List all the square numbers that are factors of 36. A radicand is the number under the radical symbol. A radical expression is in simplest form if: No radicands have. No radicands contain. No radicals appear in the. Product Property of Square Roots In words: For any nonnegative real numbers a and b, the square root of a*b is equal to the square root of a times the square root of b. In symbols: If and, then. Example: Simplifying Square Roots Directions: Simplify the following square roots Find the greatest square number that is a factor of the radicand

4 Multiplying Square Roots Directions: Simplify the following expressions Simplifying Square Roots with Variables Helpful Tips: Directions: Simplify the following expressions

5 Quotient Property of Square Roots In words: For any real numbers a and b where and, the square root of is equal to the square root of a divided by the square root of b. In symbols: Warm Up: Write down any questions you had from yesterday s intro worksheet here. Rationalizing Denominators We can rationalize, or get rid of, radicals in the denominator of a fraction by multiplying by. Directions: Simplify the following expressions. 1. How to Rationalize a Denominator ~can t have a radical in the denominator!~ Key: Multiply by a creative form of 1 2. x x x x x 3. 4.

6 Binomials of the form and are called conjugates. The product of conjugates is a rational number (no radicals!). Directions: Simplify the following expressions. 1. How to Rationalize a Denominator ~can t have a radical in the denominator!~ Key: Multiply by a creative form of 1 2. x 1 1 x x x x When fighting a fire, the velocity (v) of water being pumped into the air is modeled by the function, where h represents the maximum height of the water and g represents the acceleration due to gravity (32 ft/s 2 ). a. Solve the function for h (get h by itself). b. The Hollowville Fire Dept. needs a pump that will propel water 80 feet into the air. Will a pump with a velocity of 70 ft/s meet their needs? Explain. c. The Jackson Fire Dept. must purchase a pump that will propel water 90 feet into the air. Will a pump with a velocity of 77 ft/s meet the fire department s needs? Explain.

7 6.2.2 Completing the Square What number do we need to add to both sides to make the left hand side a perfect square? = How to Complete the Square: Examples: 1. Find the value of c that makes a perfect square. Factor. 1. Find. 2. the result from Step the result from Step Factor using the form x b 2 a 2. Find the value of c that makes a perfect square. Factor. Solving Equations by Completing the Square Examples: 1. Steps: 1. Isolate x and bx terms 2. Complete the square (see above) 3. Take square root of both sides 4. Separate solutions

8 2. 3. One Helpful Use: Complete the Square to Change into Vertex Form In the examples above, you would have to use the equation to find the axis of symmetry, then substitute that back in to the equation to find the vertex. By completing the square, you change the equation into vertex form, which makes finding the vertex much easier. Go back and identify the vertex of each of the equations on the last page. What if Examples: 1. 2.

9 3. Application: Collin is building a deck on the back of his family s house. He has enough lumber for the deck to be 144 square feet. The length should be 10 more than the width. What should the dimensions of the deck be? The Projectile Motion Model Projectile motion is always modeled by equations of the form ( ). -16 is the gravitational constant in ft/s 2. The variable is the initial velocity and is the initial height. (Note: You may also see the equation with a = -10 or -9.8, the gravitational constant in m/s 2.) Example: In competitions, skateboarders launch themselves from a half pipe into the air to perform tricks. If the height of the half pipe is 12 feet and a skateboarder launches into the air at a rate of 20 ft/s, how long will it take the skateboarder to reach a height of 25 feet?

10 6.2.3 Quadratic Formula The solutions of a quadratic equation, where, are given by the Quadratic Formula: x b ± b a ac Ask Ms. Abels for an Extra Credit assignment if you re wondering, How on earth did someone come up with that formula?!? Examples: It is important to note that the solutions of quadratic equations are not always integers. Directions: Solve using the Quadratic Formula. Simplify your answers as much as possible, but do not round

11 3. 4. Sometimes, the solutions of quadratic equations do not exist in terms of real numbers. Examples: Working Backwards: If you know the solution to a quadratic equation, you can work backwards to figure out what the quadratic equation was. Examples: 1. ± 2. ± 3. ±

12 Summary: How to Solve Quadratic Equations Method Graphing When to Use It Factoring Completing the Square Quadratic Formula

13 The Discriminant In the Quadratic Formula, the expression under the radical sign is called the discriminant and can be used to determine the number of real solutions of a quadratic equation. But how?? Solve the following three problems using the quadratic formula: Now let s see how the discriminant relates to the number of solutions: Equation Discriminant Graph # of Real Solutions

14 Using the Discriminant Directions: State the value of the discriminant and the number of real solutions of the equation Application: The equation models the distance in feet it takes a car traveling at a speed of miles per hour to come to a complete stop. If Hannah s car stopped after 250 feet on a highway with a speed limit of 65 miles per hour, was she speeding? Explain your reasoning.

15 6.2.4 Successive Differences Identifying Functions So far this year, we have studied linear functions, exponential functions, and quadratic functions. Different scenarios are modeled best by different kinds of functions. For example, we modeled the spread of a zombie virus with exponential functions, projectile motion with quadratic functions, and the cost of pizza toppings with linear functions. One way to identify which function type is the best fit for a set of data is to graph it. Directions: Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function. 1. x y X y ¼ ½ x y x y

16 Don t feel like graphing? Use successive differences! First differences are the. Second differences are the. Example: x y How to identify functions using successive differences: 1. Linear: 2. Quadratic: 3. Exponential: Directions: Identify whether the data represents a linear, quadratic, or exponential function. x y x y x y x y

17 Comparing Linear, Quadratic, and Exponential Functions Three graphs are shown at right. Their equations are as follows: ( ) ( ) ( ) ( ) 1. When, which function has the greatest value? 2. When, which function has the greatest value? 3. When, which function will have the greatest value? How do you know? Applications 1. The table shows the number of children enrolled in a beginner s karate class for 4 consecutive years. Determine which model best represents the data. Then write a function that models that data. Time (years) Number enrolled A scientist estimates that a bacteria culture with an initial population of 12 will triple every hour. a. Make a table that shows the bacteria population for the first 4 hours. b. Which kind of model best represents the data? c. Write a function that models the data. d. How many bacteria will there be after 8 hours?