GRAPHING ABSOLUTE VALUE FUNCTIONS WORKSHEET

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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

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.

Graph each equation.

Problem 1 :

y = |x - 2| - 4

Solution

Problem 2 :

y = |x + 1|

Solution

Problem 3 :

y = |x| + 1

Solution

Problem 4 :

y = |x| + 2

Solution

Problem 5 :

y = |x + 2|

Solution

Problem 6 :

y = |x + 1| + 3

Solution

Problem 7 :

y = - |x - 2| - 2

Solution

Problem 8 :

y = - |x + 1| + 4

Solution

Problem 9 :

y = -|x + 4| + 2

Solution

Problem 10 :

y = -|x - 1| + 1

Solution

Problem 11 :

y = - |x - 2| + 4

Solution

Problem 12 :

y = -|x - 1| - 1

Solution

Answer Key

1) opens upward

Vertex (2, -4)

x- Intercepts = (6, 0) and (-2, 0)

y- Intercept = (0, -2)

abs-value-s1

2) opens upward

Vertex (h, k) = (-1, 0)

x- Intercept = (-1, 0)

y- Intercept = (0, 1)

abs-value-s2

3) a > 0, opens upward

Vertex (h, k) = (0, 1)

|x| = -1

y- Intercept = (0, 1)

abs-value-s3

4) a > 0, opens upward

Vertex (h, k) = (0, 2)

|x| = -2

y- Intercept = (0, 2)

abs-value-s4

5) a > 0, opens upward

Vertex (h, k) = (-2, 0)

x- Intercept = (-2, 0)

y- Intercept = (0, 2)

abs-value-s5

6) a > 0, opens upward

Vertex (h, k) = (-1, 3)

|x + 1| = -3

y- Intercept = (0, 4)

abs-value-s6

7) a < 0, opens downward

Vertex (h, k) = (2, -2)

-|x - 2| = 2

y- Intercept = (0, -4)

abs-value-s7

8) a < 0, opens downward

Vertex (h, k) = (-1, 4)

x- Intercept = (3, 0) and (-5, 0)

y- Intercept = (0, 3)

abs-value-s8

9) a < 0, opens downward

Vertex (h, k) = (-4, 2)

x- Intercepts = (-2, 0) and (-6, 0)

y- Intercept = (0, -2)

abs-value-s9

10) a < 0, opens downward

Vertex (h, k) = (1, 1)

x- Intercepts = (2, 0) and (0, 0)

y- Intercept = (0, 0)

abs-value-s10

11) a < 0, opens downward

Vertex => (2, 4)

x- Intercepts = (6, 0) and (-2, 0)

y- Intercept = (0, 2)

abs-value-s11

12) a < 0, opens downward

Vertex (h, k) = (1, -1)

-|x - 1| = 1

y- Intercept = (0, -2)

abs-value-s12

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

solving-absolute-value-function-graphing-q2.png

Solution

Answer Key

1)

i) Solution is (1, 3).

absofxplus2p1

ii) |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

(1, 3) is the point of intersection.

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

3) |g - 1863|  = 4

4)  30 and 150 are solutions.

5)

Option a :

x = -2 and x = 5

Option b :

-2 < x < 5

Option c :

(-∞, -2) (5, ∞)

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