# Lesson 15

Vertex Form

### Problem 1

Select all of the quadratic expressions in vertex form.

A:

$$(x-2)^2 + 1$$

B:

$$x^2 - 4$$

C:

$$x(x+1)$$

D:

$$(x+3)^2$$

E:

$$(x-4)^2 + 6$$

### Solution

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### Problem 2

Here are two equations. One defines function $$m$$ and the other defines function $$p$$.

$$m(x)=x(x+6)$$

$$p(x)=(x+3)^2-9$$

1. Show that the expressions defining $$m$$ and $$p$$ are equivalent.
2. What is the vertex of the graph of $$m$$? Explain how you know.
3. What are the $$x$$-intercepts of the graph of $$p$$? Explain how you know.

### Solution

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### Problem 3

Which equation is represented by the graph?

A:

$$y=(x-1)^2+3$$

B:

$$y=(x-3)^2+1$$

C:

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

D:

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

### Solution

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### Problem 4

For each equation, write the coordinates of the vertex of the graph that represents the equation.

1. $$y=(x-3)^2 + 5$$
2. $$y=(x+7)^2 + 3$$
3. $$y=(x-4)^2$$
4. $$y=x^2 - 1$$
5. $$y=2(x+1)^2-5$$
6. $$y=\text-2(x+1)^2-5$$

### Solution

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### Problem 5

For each function, write the coordinates of the vertex of its graph and tell whether the graph opens up or down.

function coordinates of vertex graph opens up or down?
$$f(x)=(x-4)^2-5$$
$$g(x)=\text-x^2+5$$
$$h(x)=2(x+1)^2-4$$

### Solution

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### Problem 6

Here is a graph that represents $$y = x^2$$.

1. Describe what would happen to the graph if the original equation were modified as follows:

1. $$y=\text-x^2$$
2. $$y=3x^2$$
3. $$y=x^2 + 6$$
2. Sketch the graph of the equation $$y=\text-3x^2 + 6$$ on the same coordinate plane as $$y=x^2$$.

### Solution

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(From Unit 6, Lesson 12.)

### Problem 7

Noah is going to put \$2,000 in a savings account. He plans on putting the money in an account and leaving it there for 5 years. He can put the money in an account that pays 1% interest monthly, an account that pays 6% interest every six months, or an account that pays 12% interest annually.

Which account will give him the most money in his account at the end of the 5 years?

### Solution

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(From Unit 5, Lesson 16.)

### Problem 8

Here are four graphs. Match each graph with a quadratic equation that it represents.

### Solution

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(From Unit 6, Lesson 12.)

### Problem 9

The table shows some input and output values of function $$f$$. Describe a possible rule for the function by using words or by writing an equation.

$$x$$ $$f(x)$$
-3 -8
0 -2
4 6
10 18

### Solution

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(From Unit 4, Lesson 5.)