February 8 th February 12 th. Unit 6: Polynomials & Introduction to Quadratics

Transcription

1 Algebra I February 8 th February 12 th Unit 6: Polynomials & Introduction to Quadratics

2 Jump Start 1) Use the elimination method to solve the system of equations below. x + y = 2 3x + y = 8 2) Solve: =0 Graphs of Quadratic Functions Example 1: Below are some examples of curves found in architecture around the world. Some of these might be represented by graphs of quadratic functions. What are the key features these curves have in common with a graph of a quadratic function? St. Louis Arch Bellos Falls Arch Bridge Arch of Constantine Roman Aqueduct 1

3 Example 2: Use the graphs of quadratic functions (Graph A and Graph B) to fill in the table and answer the questions on the following page. ( ) 1 8 Graph A f(x) = (x-2) ( ) -5-4 Graph B f(x) = -(x+2)

4 Graph A Graph B 1 -Intercepts 2 Vertex 3 Sign of the Leading Coefficient 4 Vertex Represents a Minimum or Maximum? 5 Line of Symmetry On what intervals of the domain is the function depicted by the graph increasing? On what intervals of the domain is the function depicted by the graph increasing? 6 Increasing and Decreasing Intervals On what intervals of the domain is the function depicted by the graph decreasing? On what intervals of the domain is the function depicted by the graph decreasing? 3

5 Leading coefficient is positive: Leading coefficient is negative: 4

6 Independent Practice 1) Consider the following key features discussed in this lesson for the four graphs of quadratic functions below: -intercepts, -intercept, line of symmetry, vertex, and intervals when the functions are increasing and decreasing. Graph A Graph B x-intercepts y-intercept line of symmetry vertex increasing & decreasing Graph A Graph B 2) Consider the equation = Given this quadratic equation, can you find the point(s) where the graph crosses the -axis? Y-intercept? Vertex? 5

7 3) Given the equation, ( )= 5 24 find the x-intercepts, y-intercept, and vertex. 4) Given the equation, ( )= 5( 2)( 3) find the x-intercepts, y-intercept, and vertex. 5) A rocket is launched from a cliff. The relationship between the height, h, in feet, of the rocket and the time,, in seconds, since its launch can be represented by the following function: a. When will the rocket hit the ground? h( )= b. When will the rocket reach its maximum height? c. What is the maximum height the rocket reaches? d. At what height was the rocket launched? 6

8 Jump Start A toy company is manufacturing a new toy and trying to decide on a price that will result in a maximum profit. The graph below represents profit ( ) generated by each price of a toy ( ). = ( ) a. If the company wants to make a maximum profit, what should the price of a new toy be? b. What is the minimum price of a toy that will produce profit for the company? Explain your answer. c. Estimate the value of (0), and explain what the value means in the problem. d. If the company wants to make a profit of $137, for how much should the toy be sold? e. Find the domain that will only result in a profit for the company, and find its corresponding range of profit. 7

9 Unit 6 Test Review Questions 1) What is the area of the rectangle? 3x 2 2x + 6 2) Jeff has a square play pen for his dog with a length of s. He plans to make it into a rectangular play pen by extending one set of sides by 4 feet. Write an equation that represents the new area, N, of the play pen? 3) Which expression is equivalent to t 2 36? A (t 6)(t 6) B (t + 6)(t 6) C (t 12)(t 3) D (t 12)(t + 3) 4) What are the zeros of the function ( )= 13 30? *Remember- zeros = x-intercepts! 5) A student is asked to solve the equation 4(3 1) 17=83. The student s solution to the problem starts as 4(3 1) =100 (3 1) =25 Finish solving the equation. 8

10 6) A landscaper is creating a rectangular flower bed such that the width is half of the length. The area of the flower bed is 34 square feet. Write and solve an equation to determine the width of the flower bed, to the nearest tenth of a foot. 7) Solve algebraically for x : =2 8) Solve the equation: 2( 3) =12 9

11 9) The goal post is 10 feet high and 45 yards away from the kick. Will the ball be high enough to pass over the goal post? Justify your answer. [Hint: There are 3 feet in 1 yard] 10

12 Week 23 Homework 11