# 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 \(f(x)=(x-7)(x+3)\).

- Without graphing, identify the \(x\)-intercepts of the graph of \(f\). Explain how you know.
- 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 \((x-2)(2x+1)\)?

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

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

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

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

### Problem 3

Here is a graph that represents a quadratic function.

Which expression could define this function?

\((x+3)(x+1)\)

\((x+3)(x-1)\)

\((x-3)(x+1)\)

\((x-3)(x-1)\)

### Problem 4

- What is the \(y\)-intercept of the graph of the equation \(y = x^2 - 5x + 4\)?
- 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?

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

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

\(\dfrac{20,000 - 500p}{p}\)

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

### Problem 7

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

- \((x-3)(x-6)\)
- \((x-4)^2\)
- \((2x+3)(x-4)\)
- \((4x-1)(3x-7)\)

### Problem 8

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

- Is the expression equivalent to \(x^2+x+30\)? Explain how you know.
- Is the expression \(30+x-x^2\) in standard form? Explain how you know.

### 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.

### 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 \(g\)?

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

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

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

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