# Lesson 10

Graphs of Functions in Standard and Factored Forms

• Let’s find out what quadratic expressions in standard and factored forms can reveal about the properties of their graphs.

### Problem 1

A quadratic function $$f$$ is defined by $$f(x)=(x-7)(x+3)$$.

1. Without graphing, identify the $$x$$-intercepts of the graph of $$f$$. Explain how you know.
2. Expand $$(x-7)(x+3)$$ and use the expanded form to identify the $$y$$-intercept of the graph of $$f$$.

### Problem 2

What are the $$x$$-intercepts of the graph of the function defined by $$(x-2)(2x+1)$$?

A:

$$(2,0)$$ and $$(\text-1,0)$$

B:

$$(2,0)$$ and $$\left(\text-\frac12,0\right)$$

C:

$$(\text-2,0)$$ and $$(1,0)$$

D:

$$(\text-2,0)$$ and $$(\frac12,0)$$

### Problem 3

Here is a graph that represents a quadratic function.

Which expression could define this function?

A:

$$(x+3)(x+1)$$

B:

$$(x+3)(x-1)$$

C:

$$(x-3)(x+1)$$

D:

$$(x-3)(x-1)$$

### Problem 4

1. What is the $$y$$-intercept of the graph of the equation $$y = x^2 - 5x + 4$$?
2. An equivalent way to write this equation is $$y = (x-4)(x-1)$$. What are the $$x$$-intercepts of this equation’s graph?

### Problem 5

Noah said that if we graph $$y=(x-1)(x+6)$$, the $$x$$-intercepts will be at $$(1,0)$$ and $$(\text-6,0)$$. Explain how you can determine, without graphing, whether Noah is correct.

### Problem 6

A company sells a video game. If the price of the game in dollars is $$p$$ the company estimates that it will sell $$20,\!000 - 500p$$ games.

Which expression represents the revenue in dollars from selling games if the game is priced at $$p$$ dollars?

A:

$$(20,\!000 - 500p) + p$$

B:

$$(20,\!000 - 500p) - p$$

C:

$$\dfrac{20,000 - 500p}{p}$$

D:

$$(20,\!000 - 500p) \boldcdot p$$

(From Unit 6, Lesson 7.)

### Problem 7

Write each quadratic expression in standard form. Draw a diagram if needed.

1. $$(x-3)(x-6)$$
2. $$(x-4)^2$$
3. $$(2x+3)(x-4)$$
4. $$(4x-1)(3x-7)$$
(From Unit 6, Lesson 9.)

### Problem 8

Consider the expression $$(5+x)(6-x)$$.

1. Is the expression equivalent to $$x^2+x+30$$? Explain how you know.
2. Is the expression $$30+x-x^2$$ in standard form? Explain how you know.
(From Unit 6, Lesson 9.)

### Problem 9

Here are graphs of the functions $$f$$ and $$g$$ given by $$f(x) = 100 \boldcdot \left(\frac{3}{5}\right)^x$$ and $$g(x) = 100 \boldcdot \left(\frac{2}{5}\right)^x$$.

Which graph corresponds to $$f$$ and which graph corresponds to $$g$$? Explain how you know.

(From Unit 5, Lesson 12.)

### Problem 10

Here are graphs of two functions $$f$$ and $$g$$.

An equation defining $$f$$ is $$f(x) = 100 \boldcdot 2^x$$.

Which of these could be an equation defining the function $$g$$?

A:

$$g(x) = 25 \boldcdot 3^x$$

B:

$$g(x) = 50 \boldcdot (1.5)^x$$

C:

$$g(x) = 100 \boldcdot 3^x$$

D:

$$g(x) = 200 \boldcdot (1.5)^x$$

(From Unit 5, Lesson 13.)