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 \((x-2)(2x+1)\)?


\((2,0)\) 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?

A curve in an x y plane, origin O, with grid.








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?


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


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


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


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

(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.

Graph of two lines.
(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 \(g\)?

Graph of two increasing exponential functions, xy-plane, origin O.

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


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


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


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

(From Unit 5, Lesson 13.)