GRAPHING ABSOLUTE VALUE FUNCTIONS

To graph absolute value function, we have to find the following characteristics of the given function.

(i) Find vertex

(ii)  x - intercepts (roots, zeroes, solutions) and y - intercept

(iii) Slope and Reflections (or) Direction of opening

(iv)  Domain and Range

(v)  Increasing/decreasing interval

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 :

Finding vertex :

y = 3|x - 3|

Comparing with y = a|x - h| + k

y = 3|x - 3| + 0

Vertex is at (3, 0).

x and y-intercepts :

x-intercept, put y = 0

3|x - 3| = 0

Since we have zero on the right side, don't have to decompose it into two branches.

|x - 3| = 0

x = 3

x-intercept is (3, 0).

y-intercept, put x = 0

y = 3|0 - 3|

y = 3(3)

y = 9

y-intercept is at (0, 9).

Slope : 

a = 3

The curve will open up.

Domain and range :

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

Increasing and Decreasing :

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

Problem 2 :

Graph

f(x) = |x − 4| − 1

Determine when the function is positive, negative, increasing, or decreasing. Then describe the end behavior of the function.

Solution :

Comparing with y = a|x - h| + k

• Vertex is at (4, -1)
• Opening up.

x - intercept :

Put y = 0

|x − 4| − 1 = 0

|x - 4| = 1

x - 4 = 1 and x - 4 = -1

x = 1 + 4 and x = -1 + 4

x = 5 and x = 3

(5, 0) and (3, 0)

y - intercept :

Put x = 0

y = |0 − 4| − 1

y = 4 - 1

y = 3

(0, 3)

Positive :

When x < 3 and x > 5

Negative :

When 3 < x < 5

Increasing or decreasing :

Decreasing on (-∞, 4)

Increasing on (4, ∞)

End behavior :

When x -> -∞ y --> ∞

When x -> ∞ y --> ∞

Problem 3 :

A function g is increasing when x < 2, decreasing when x > 2, and has a range of (−∞, −2). Use the given values to complete the function. Do not use any value more than once.

-2      0       2     - 1

Solution :

The absolute value function should be in the form

y = a|x - h| + k

From the given range, the function opens down.

• When x < 2, the absolute value function is increasing.
• When x > 2, the absolute value function is decreasing.
• The value of k = 0

y = -1|x - 2| + 0

So, the required function is y = |x - 2|

Problem 4 :

Graph each absolute value function f with the given characteristics.

a) f has a range of (−∞, 1), and a graph that is symmetric about the line x = −2 and has a y-intercept of −5.

b) f is positive over the intervals (−∞, 0) and (4, ∞), negative over the interval (0, 4), and the minimum value is −4.

Solution :

Because the graph is symmetric about x = −2, the x-value of the vertex is −2. Because the range is (−∞, 1), the y-value of the vertex is 1.

Plot the vertex (−2, 1). Because the y-intercept is − 5, plot the point (0, − 5) and its reflection in the line of symmetry, (−4, −5). Then draw the graph.