Changing the Vertex

ID: rukas-pohuz
Illustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2

32 questions

Changing the Vertex

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Problem 1

Here the graph of quadratic function ﻿$f$﻿.

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

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

1) Do you agree with either of them?

True or false? Write below.

Problem 2

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

3) How do the 3 graphs compare?

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

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

Problem 3

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

a)

﻿$y$﻿=﻿$x^{2}$﻿+2﻿$x$﻿-6

b)

﻿$y$﻿=(﻿$x$﻿+3﻿$)^{2}$﻿+2﻿$x$﻿-3

c)

﻿$y$﻿=(﻿$x$﻿+3﻿$)^{2}$﻿+2(﻿$x$﻿+3)

d)

﻿$y$﻿=(﻿$x$﻿+3﻿$)^{2}$﻿+2(﻿$x$﻿+3)-3

Problem 4

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

a) ﻿$y = x^2 \quad \quad\quad$﻿ b) ﻿$y = (x - 1)^2 \quad \quad$﻿ c) ﻿$y = (x - 3)^2 + 2 \quad \quad$﻿ d) ﻿$y = 2(x - 4)^2 - 5 \quad \quad\quad$﻿ e) ﻿$y = 0.5(x + 2)^2 + 6$﻿

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

Problem 5

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

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

Show Work

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

Show Work

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

Problem 6

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

Show Work

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

Show Work

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

Show Work

14) Sketch the graph of ﻿$f$﻿.

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.

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. 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 ﻿$h$﻿ is defined by ﻿$h(x) = 5x + 7$﻿ and function ﻿$k$﻿ is defined by ﻿$k(x) = (1.005)^x$﻿.

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

23) ﻿$x \ = \ 1$﻿

Show Work

24) ﻿$x \ = \ 10$﻿

Show Work

25) ﻿$x \ = \ 50$﻿

Show Work

26) ﻿$x \ = \ 100$﻿

Show Work

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

27) ﻿$x \ = \ 1$﻿

Show Work

28) ﻿$x \ = \ 10$﻿

Show Work

29) ﻿$x \ = \ 50$﻿

Show Work

30) ﻿$x \ = \ 100$﻿

Show Work

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