Changing the Vertex

ID: rukas-pohuz
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
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32 questions

Changing the Vertex

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Problem 1

Here the graph of quadratic function ff.

Andre uses the expression (x5)2+7(x - 5)^2 + 7 to define ff.

Noah uses the expression (x+5)27(x + 5)^2 -7 to define ff.

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1) Do you agree with either of them?

True or false? Write below.

2) Explain your reasoning.

Problem 2

Here are the graphs of y=x2y = x^2, y=x25y = x^2 - 5, and y=(x+2)28y = (x + 2)^2 - 8.

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3) How do the 3 graphs compare?

4) How does the -5 in x25x^2 - 5 affect the graph?

5) How does the +2 and the -8 in (x+2)28(x + 2)^2 - 8 affect the graph?

Problem 3

6) Which equation represents the graph of y=x2+2x3y = x^2 + 2x - 3 moved 3 units to the left?

a)

yy=x2x^{2}+2xx-6

b)

yy=(xx+3)2)^{2}+2xx-3

c)

yy=(xx+3)2)^{2}+2(xx+3)

d)

yy=(xx+3)2)^{2}+2(xx+3)-3

Problem 4

7) Select all\textbf{all} the equations with a graph whose vertex has both\textit{both} a positive xx- and a positive yy-coordinate. Write each corresponding letter in the answer box and separate letters with commas.

a) y=x2y = x^2 \quad \quad\quad b) y=(x1)2y = (x - 1)^2 \quad \quad c) y=(x3)2+2y = (x - 3)^2 + 2 \quad \quad d) y=2(x4)25y = 2(x - 4)^2 - 5 \quad \quad\quad e) y=0.5(x+2)2+6y = 0.5(x + 2)^2 + 6

f) y=(x4)2+3y = -(x - 4)^2 + 3 \quad \quad g) y=2(x3)2+1y = -2(x - 3)^2 + 1 \quad \quad

Problem 5

The height in feet of a soccer ball is modeled by the equation g(t)=2+50t16t2g(t) = 2 + 50t - 16t^2, where time tt is measured in seconds after it was kicked.

8) How far above the ground was the ball when kicked?

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9) What was the initial upward velocity of the ball?

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10) Why is the coefficient of the squared term negative?

Problem 6

11) What is the vertex of the graph of the function ff defined by f(x)=(x3)2+6f(x) = -(x - 3)^2 + 6?

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12) Identify the yy-intercept on the graph of this function.

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13) Identify one other point on the graph of this function.

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14) Sketch the graph of ff.

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Problem 7

At 6:00 a.m., Lin began hiking. At noon, she had hiked 12 miles. At 4:00 p.m., Lin finished hiking with a total trip of 26 miles.

15) During which time interval was Lin hiking faster?

16) Explain how you know.

Problem 8

Kiran bought a smoothie every day for a week. Smoothies cost $3 each. The amount of money he spends, in dollars, is a function of the number of days of buying smoothies.

17) Sketch a graph of this function. Be sure to label the axes.

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18) Describe the domain and range of this function.

Problem 9

A deposit of $500 has been made in an interest-bearing account. No withdrawals or other deposits (aside from earned interest) are made for 5 years.

Write an expression to represent the account balance for each of the following situations.

19) 6.5% interest calculated monthly

20) 6.5% interest calculated every two months

21) 6.5% interest calculated quarterly

22) 6.5% interest calculated semi-annually

Problem 10

Function hh is defined by h(x)=5x+7h(x) = 5x + 7 and function kk is defined by k(x)=(1.005)xk(x) = (1.005)^x.

Find h(x)h(x) for each of the following values of xx. When necessary, round to 2 decimal places.

23) x = 1x \ = \ 1

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24) x = 10x \ = \ 10

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25) x = 50x \ = \ 50

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26) x = 100x \ = \ 100

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Find k(x)k(x) for each of the following values of xx. When necessary, round to 2 decimal places.

27) x = 1x \ = \ 1

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28) x = 10x \ = \ 10

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29) x = 50x \ = \ 50

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30) x = 100x \ = \ 100

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31) Which function do you think eventually\textit{eventually} grows faster?

32) Explain your reasoning.