# Lesson 17

Changing the Vertex

- Let’s write new quadratic equations in vertex form to produce certain graphs.

### 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 \(f\).

Do you agree with either of them? Explain your reasoning.

### Problem 2

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

- How do the 3 graphs compare?
- Compare the graphs of \(y=x^2\) and \(y = x^2-5\). What role does the -5 play in the comparison?
- Compare the graphs of \(y=x^2\) and \(y=(x+2)^2-8\). What role does the +2 play in the comparison?

### Problem 3

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

\(y=x^2+2x-6\)

\(y=(x+3)^2+2x-3\)

\(y=(x+3)^2+2(x+3)\)

\(y=(x+3)^2+2(x+3)-3\)

### Problem 4

Select **all** the equations with a graph whose vertex has *both* a positive \(x\)- and a positive \(y\)-coordinate.

\(y=x^2\)

\(y=(x-1)^2\)

\(y=(x-3)^2+2\)

\(y=2(x-4)^2-5\)

\(y=0.5(x+2)^2+6\)

\(y=\text-(x-4)^2+3\)

\(y=\text-2(x-3)^2+1\)

### 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.

- How far above the ground was the ball when kicked?
- What was the initial upward velocity of the ball?
- Why is the coefficient of the squared term negative?

### Problem 6

- What is the vertex of the graph of the function \(f\) defined by \(f(x)=\text-(x-3)^2+6\)?
- Identify the \(y\)-intercept and one other point on the graph of this function.
- 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.

During which time interval was Lin hiking faster? 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.

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

- 6.5% annual interest calculated monthly
- 6.5% annual interest calculated every two months
- 6.5% annual interest calculated quarterly
- 6.5% annual interest calculated semi-annually

### Problem 10

*Technology required*. Function \(h\) is defined by \(h(x) = 5x+7\) and function \(k\) is defined by \(k(x) = (1.005)^x\).

- Complete the table with values of \(h(x)\) and \(k(x)\). When necessary, round to 2 decimal places.
- Which function do you think
*eventually*grows faster? Explain your reasoning. - Use graphing technology to verify your answer to the previous question.

\(x\) | \(h(x)\) | \(k(x)\) |
---|---|---|

1 | ||

10 | ||

50 | ||

100 |