Lesson 10

Graphs of Functions in Standard and Factored Forms

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

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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

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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

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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

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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

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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

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(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

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(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

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(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

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(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

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(From Unit 5, Lesson 13.)