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Algebra II : Graphing Absolute Value Functions

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Example Questions

Example Question #1 : Graphing Absolute Value Functions

Axes_1

Refer to the above figure.

Which of the following functions is graphed?

Possible Answers:

g(x) = - |x| + 6

g(x) = |x| - 6

g(x) =- |x + 6|

g(x) = - |x|- 6

g(x) =- |x - 6|

Correct answer:

g(x) = - |x| + 6

Explanation:

Below is the graph of f(x) = |x| :

Axes_1

The given graph is the graph of reflected in the -axis, then translated up 6 units. This graph is

g (x) = -f(x ) + k , where k = 6 .

The function graphed is therefore

g (x) = -f(x ) + k

g (x) = -f(x ) + 6

g (x) = - |x|+ 6

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Example Question #2 : Graphing Absolute Value Functions

Axes_1

Refer to the above figure.

Which of the following functions is graphed?

Possible Answers:

g(x) = - |x +2|

g(x) =2 - |x |

g(x) = -2 - |x |

g(x) = - |x -2|

g(x) = |x | -2

Correct answer:

g(x) = - |x +2|

Explanation:

Below is the graph of f(x) = |x| :

Axes_1

The given graph is the graph of reflected in the -axis, then translated left 2 units (or, equivalently, right units. This graph is

g (x) = -f(x-h) , where h = -2 .

The function graphed is therefore

g (x) = -f(x- (-2))

g (x) = -f(x+2)

g(x) = - |x +2|

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Example Question #3 : Graphing Absolute Value Functions

Axes_1

Refer to the above figure.

Which of the following functions is graphed?

Possible Answers:

The correct answer is not given among the other responses.

g(x)= | x-7 | - 2

g(x)= | x-7 | + 2

g(x)= | x+7 | + 2

g(x)= | x+7 | - 2

Correct answer:

g(x)= | x+7 | - 2

Explanation:

Below is the graph of f(x) = |x| :

Axes_1

The given graph is the graph of translated by moving the graph 7 units left (that is, unit right) and 2 units down (that is, units up)

The function graphed is therefore

g (x) = f(x-h) + k where h = -7, k= -2 . That is,

g (x) = f(x-(-7)) + (-2)

g (x) = f(x+7) -2

g(x)= | x+7 | - 2

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Example Question #3 : Graphing Absolute Value Functions

Screen_shot_2014-12-24_at_3.03.30_pm

What is the equation of the above function?

Possible Answers:

y=|2x+2|-3

y=|x+2|+3

$y=|\frac{x}{2}+2|-3$

y=|x+2|-3

y=|2(x+2)|-3

Correct answer:

y=|2(x+2)|-3

Explanation:

The formula of an absolute value function is y=|m(x-a)|+b where m is the slope, a is the horizontal shift and b is the vertical shift. The slope can be found with any two adjacent integer points, e.g. (0,1) and (1,3) , and plugging them into the slope formula, $m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ , yielding m=2 . The vertical and horizontal shifts are determined by where the crux of the absolute value function is. In this case, at (-2,-3) , and those are your a and b, respectively.

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Example Question #4 : Graphing Absolute Value Functions

Give the vertex of the graph of the function f(x) = |x + 6| - 7 .

Possible Answers:

(-6, -7)

(6, -7)

(-6, 7)

None of the other choices gives the correct response.

(6, 7)

Correct answer:

(-6, -7)

Explanation:

Let g(x) = |x|

The graph of this basic absolute value function is a "V"-shaped graph with a vertex at the origin, or the point with coordinates (0,0) . In terms of g(x) ,

f(x) = g(x + 6 )- 7

The graph of this function can be formed by shifting the graph of g(x) left 6 units ( ) and down 7 units (). The vertex is therefore located at (-6, -7) .

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Example Question #2 : Graphing Absolute Value Functions

Give the vertex of the graph of the function f(x) = 10 - |x+ 10| .

Possible Answers:

(-10, 10)

(10, -10)

None of the other choices gives the correct response.

(10,10)

(-10, -10)

Correct answer:

(-10, 10)

Explanation:

Let g(x) = |x|

The graph of this basic absolute value function is a "V"-shaped graph with a vertex at the origin, or the point with coordinates (0,0) . In terms of g(x) ,

f(x) = 10 - g (x+ 10) ,

or, alternatively written,

f(x) = - g (x+ 10)+ 10

The graph of f(x) is the same as that of g(x) , after it shifts 10 units left ( g(x+10) ), it flips vertically (negative symbol), and it shifts up 10 units (the second +10 ). The flip does not affect the position of the vertex, but the shifts do; the vertex of the graph of f(x) is at (-10, 10) .

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Example Question #7 : Graphing Absolute Value Functions

Which of the following absolute value functions is represented by the following graph?

Possible Answers:

$f(x)=2\left | x-3\right |+1$

$f(x)=3\left | x-3\right |+1$

$f(x)=2\left | x+3\right |+1$

$f(x)=3\left | x+3\right |+1$

The equation cannot be determined from the graph.

Correct answer:

$f(x)=2\left | x-3\right |+1$

Explanation:

The equation can be determined from the graph by following the rules of transformations; the base equation is:

$f(x)=\left | x\right |$

The graph of this base equation is:

When we compare our graph to the base equation graph, we see that it has been shifted right 3 units, up 1 unit, and our graph has been stretched vertically by a factor of 2. Following the rules of transformations, the equation for our graph is written as:

$f(x)=2\left | x-3\right |+1$

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Algebra II : Graphing Absolute Value Functions

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Graphing Absolute Value Functions

Example Question #2 : Graphing Absolute Value Functions

Example Question #3 : Graphing Absolute Value Functions

Example Question #3 : Graphing Absolute Value Functions

Example Question #4 : Graphing Absolute Value Functions

Example Question #2 : Graphing Absolute Value Functions

Example Question #7 : Graphing Absolute Value Functions

All Algebra II Resources

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