lesson 5: investigating quadratic functions in the...

ALGEBRA I

Lesson 5: Investigating Quadratic Functions in the Standard Form, 𝒇𝒇(𝒙𝒙) = 𝒂𝒂𝒙𝒙𝟐𝟐 + 𝒃𝒃𝒙𝒙 + 𝒄𝒄

Opening Exercise

1. Marshall had the equation y = (x – 2)2 + 4 and knew that he could easily find the vertex. Sarah said that he

could find the y-intercept if he rewrote his equation into standard form. Sarah got him started by rewriting

his equation as shown below. Continue Sarah’s work to show the steps Marshall would need to take to get

a y-intercept of 8 from his original equation.

y = (x – 2)2 + 4

y = (x – 2)• (x – 2) + 4

2. What is the vertex of Marshall’s quadratic?

3. Use the vertex and y-intercept to sketch a graph of

Marshall’s parabola. What could you do to get a

more accurate graph?

Lesson 5: Investigating Quadratic Functions in the Standard Form, 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥2 + 𝑏𝑏𝑥𝑥 + 𝑐𝑐

Unit 10: Introduction to Quadratics and Their Transformations

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Hart Interactive – Algebra 1 M4 Lesson 5

ALGEBRA I

Discussion 4. A. What feature(s) of this quadratic function are visible when it is presented in the standard form,

𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥2 + 𝑏𝑏𝑥𝑥 + 𝑐𝑐?

B. What feature(s) of this quadratic function are visible when it is rewritten in vertex form,

𝑓𝑓(𝑥𝑥) = 𝑎𝑎(𝑥𝑥 − ℎ)2 + 𝑘𝑘?

5. A general strategy for graphing a quadratic function from the standard form is:

________________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

6. Graph the function 𝑛𝑛(𝑥𝑥) = 𝑥𝑥2 − 6𝑥𝑥 + 5, and identify the key features.



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ALGEBRA I

Practice: Vertex Form to Standard Form

For each exercise below, rewrite the vertex form into standard form and then identify all the important

features of each quadratic.

7. Vertex form: y = (x + 6)2 – 4

Standard form: __________________________

Key Features

Vertex: ____________________

y-intercept: _________________

Does the graph open up or open

down? _____________________

Axis of symmetry: ____________

8. Vertex form: y = (x – 1 )2 – 7

Standard form: __________________________

Key Features

Vertex: ____________________

y-intercept: _________________


down? _____________________


9. Vertex form: y = – (x – 3 )2 + 2

Standard form: __________________________

Key Features

Vertex: ____________________

y-intercept: _________________


down? _____________________




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ALGEBRA I

Application Problem

10. A high school baseball player throws a ball straight up into the air for his math class. The math class was

able to determine that the relationship between the height of the ball and the time since it was thrown

could be modeled by the function ℎ(𝑡𝑡) = −16𝑡𝑡2 + 96𝑡𝑡 + 6, where 𝑡𝑡 represents the time (in seconds)

since the ball was thrown, and ℎ represents the height (in feet) of the ball above the ground.

A. The domain is _____________________ and it represents __________________________________.

B. What does the range of this function represent?

C. What is the time (t) when the ball is thrown?

D. At what height does the ball get thrown?

E. In vertex form the equation is h(t) = -16(t – 3)2 + 150. What

is the vertex of this parabola?

F. What is the maximum height that the ball reaches while in

the air? How long will the ball take to reach its maximum

height?

G. It would be difficult to tell from the equation how many

seconds it takes the ball to hit the ground. Graph the

equation in the grid at the right and make an estimate.



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ALGEBRA I

Lesson Summary

The standard form of a quadratic function is 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥2 + 𝑏𝑏𝑥𝑥 + 𝑐𝑐, where 𝑎𝑎 ≠ 0. From the standard form you can easily see that the y-intercept is at c.

A general strategy to graphing a quadratic function from the standard form:

� Look for hints in the function’s equation for general shape, direction, and 𝑦𝑦-intercept. � Use a T-chart to find more points on the graph. � Remember that a parabola is symmetric about the vertex. This can help you identify

other points you may need. � Plot the points that you know (at least three are required for a unique quadratic

function), sketch the graph of the curve that connects them, and identify the key features of the graph.



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ALGEBRA I

Homework Problem Set 1. Graph each quadratic equation given both the standard and vertex forms below.

A. 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 − 2𝑥𝑥 − 15 or

𝑓𝑓(𝑥𝑥) = (𝑥𝑥 − 1)2 − 16

B. 𝑓𝑓(𝑥𝑥) = −𝑥𝑥2 + 2𝑥𝑥 + 15 or

𝑓𝑓(𝑥𝑥) = −(𝑥𝑥 − 1)2 + 16

2. The equation in Part B of Problem 1 is the product of −1 and the equation in Part A. What effect did

multiplying the equation by −1 have on the graph?

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ALGEBRA I

3. Paige wants to start a summer lawn-mowing business. She comes up with the following profit function

that relates the total profit to the rate she charges for a lawn-mowing job:

𝑃𝑃(𝑥𝑥) = −𝑥𝑥2 + 40𝑥𝑥 − 100 or 𝑃𝑃(𝑥𝑥) = −(𝑥𝑥 − 20)2 + 300.

Both profit and her rate are measured in dollars.

A. Graph the function to help you

answer the following questions.

B. According to the function, what is her

initial cost (e.g., maintaining the

mower, buying gas, advertising)?

Explain your answer in the context of

this problem.

C. Between what two prices does she

have to charge to make a profit?

D. If she wants to make $275 profit this

summer, is this the right business

choice? Explain.

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ALGEBRA I

4. A student throws a bag of chips to her friend. Unfortunately, her friend does not catch the chips, and the

bag hits the ground. The distance from the ground (height) for the bag of chips is modeled by the

function ℎ(𝑡𝑡) = −16𝑡𝑡2 + 32𝑡𝑡 + 4 or ℎ(𝑡𝑡) = −16(𝑡𝑡 + 1)2 + 20, where ℎ is the height (distance from

the ground in feet) of the chips, and 𝑡𝑡 is the number of seconds the chips are in the air.

A. Graph ℎ.

B. From what height are the chips being thrown?

Explain how you know.

C. What is the maximum height the bag of chips

reaches while airborne? Explain how you know.

D. About how many seconds after the bag was

thrown did it hit the ground?

E. What is the average rate of change of height for

the interval from 0 to 12 second? What does that

number represent in terms of the context?

F. Based on your answer to part (e), what is the average rate of change for the interval from 1.5 to 2 sec.?

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lesson 5: investigating quadratic functions in the...

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