Lesson 10

Graphs of Functions in Standard and Factored Forms

View Student Lesson

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 \(f\).

Solution

For access, consult one of our IM Certified Partners.

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 \((\frac12,0)\)

Solution

For access, consult one of our IM Certified Partners.

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.
A:

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

B:

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

C:

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

D:

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

Solution

For access, consult one of our IM Certified Partners.

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?

Solution

For access, consult one of our IM Certified Partners.

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.

Solution

For access, consult one of our IM Certified Partners.

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:

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

Solution

For access, consult one of our IM Certified Partners.

(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. \((4x-1)(3x-7)\)

Solution

For access, consult one of our IM Certified Partners.

(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.

Solution

For access, consult one of our IM Certified Partners.

(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.

Solution

For access, consult one of our IM Certified Partners.

(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.
A:

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

B:

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

C:

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

D:

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

Solution

For access, consult one of our IM Certified Partners.

(From Unit 5, Lesson 13.)