CHARACTERISTICS OF ABSOLUTE VALUE FUNCTIONS WORKSHEET

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Find the following and graph it.

(i) Vertex

(ii) Slope

(iii) y-intercept and x-intercept

(iv) domain and range

(v) Increasing and decreasing interval.

Problem 1 :

f(x) = -3│x - 4│ + 3

Solution

Problem 2 :

f(x) = -1/2│x - 2│ + 4

Solution

Problem 3 :

f(x) = │x - 3│ - 2

Solution

Problem 4 :

f(x) = 3│x│

Solution

Problem 5 :

y = 3/5 │x│- 6

Solution

Problem 6 :

y =│x + 4│

Solution

Answer Key

Graph the following absolute value function by finding the following.

(i) Vertex

(ii) Slope

(iii) Direction of opening

(iv) x and y intercepts

(v) Domain and range

(vi) Increasing and decreasing

Problem 1 :

y = 3|x - 3|

Solution

Problem 2 :

y = -|x| + 4

Solution

Problem 3 :

y = (4/3) |x + 2| - 5

Solution

Problem 4 :

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

Solution

Problem 5 :

Match each function with its graph. Explain your reasoning.

i. f(x) = │x + 2│ − 2

ii. g(x) = −│x − 2│ + 2

iii. f(x) = −│x − 2│ − 2

iv. m(x) = │x + 2│ + 2

Solution

Answer Key

1) Vertex is at (3, 0)

Slope (a) = 3

Direction of opening = open up

x-intercept is at (3, 0)

y-intercept is at (0, 9).

All real values is domain.
Range is 3 ≤ y ≤ ∞

To the left of minimum, it is decreasing.
To the right of minimum, it is increasing.

2) Vertex is at (0, 4).

x-intercepts are (4, 0) and (-4, 0).

y-intercept is at (0, 4).

Slope (a) = -1

The curve will open down.

All real values is domain.
Range is 4 ≤ y ≤ -∞

To the left of maximum, it is increasing.
To the right of maximum, it is decreasing.

3) Vertex is at (-2, -5).

x-intercept is at (7/4, 0) and (-19/2, 0).

y-intercept (0, -7/3).

Slope (a) = 4/3

The curve will open up.

All real values is domain.
Range is 4 ≤ y ≤ -∞

To the left of minimum, it is decreasing.
To the right of minimum, it is increasing.

4) Vertex is at (3, 2).

x-intercept is at (13/3, 0) and (5/3, 0).

y-intercept is (0, -5/2).

Slope (a) = -3/2

The curve will open down.

All real values is domain.
Range is -5 ≤ y ≤ ∞

To the left of maximum, it is increasing.
To the right of maximum, it is decreasing.

5) i) Graph C is correct

ii) Graph B is correct.

iii) Graph D is correct.

iv) Graph A is correct.

Solve for x using

i) graphical method ii) an algebraical method.

Problem 1 :

|x + 2| = 2x + 1

Solution

Problem 2 :

Consider the graph, which shows y₁ = |6 - 2x| and y₂ = 18. Use the graph and use algebraic properties to solve |6 - 2x| = 18.

solving-absolute-value-function-graphing-q1

Solution

Problem 3 :

You are asked for the year of the Emancipation Proclamation in the United States on a test. The correct answer is 1863. You guessed g and you were off by 4 years. What equation’s solution gives the possible values of g?

a) |1863 - 4| = g b) |g| = 1863 - 4 c) |g - 1863| = 4 d) |g - 4| = 1863

Solution

Problem 4 :

Determine whether the number is a solution to the equation

60 = |n - 90|

a) 30 b) −30 c) 150 d) −150

Solution

Problem 5 :

Use the table below to solve each sentence.

a) |2x - 3| = 7 b) |2x - 3| < 7 c) |2x - 3| > 7

Solution

Answer Key

the solution is (1, 3).

Solving algebraically :

|x + 2| = 2x + 1

x + 2 = (2x + 1)

x + 2 = 2x + 1

x - 2x = 1 - 2

-x = -1

x = 1

(x + 2) = -(2x + 1)

x + 2 = -2x - 1

x + 2x = -1 - 2

-3x = 3

x = -1

2) the point of intersections are -6 and 12.

3) |g - 1863| = 4

4) 30 and 150 are solutions.

5) (-∞, -2) (5, ∞)

Write the equation of the graph. Then give its range as an inequality.

Problem 1 :

Problem 2 :

Problem 3 :

Problem 4 :

Problem 5 :

Problem 6 :

Problem 7 :

Describe and correct the error in graphing the function.

Solution

Problem 8 :

Describe and correct the error in graphing the function.

Solution

Answer Key

y = 1 |x +1| + 2

Range is 2 ≤ y ≤ ∞

y = -1 |x + 3| - 2

Range is -2 ≤ y ≤ -∞

y = -2 |x - 2| + 3

Range is 3 ≤ y ≤ -∞

y = (1/2) |x + 2| - 3

Range is -3 ≤ y ≤ ∞

y = (-3/4) |x - 1|

Range is 1 ≤ y ≤ -∞

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

Range is -2 ≤ y ≤ ∞

7) By observing the graph, the function opens up but the vertex is at (-1, -3) and that is the error.

8) By observing the graph, it opens up. But it should be open down and that is the error.

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CHARACTERISTICS OF ABSOLUTE VALUE FUNCTIONS WORKSHEET

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