FIND DOMAIN AND RANGE OF ABSOLUTE VALUE FUNCTION FROM GRAPH

What is domain from graph ?

The horizontal spread or the spreading over the x-axis is knowns as domain.

The vertical spread or the spreading over the y-axis is knowns as range.

For absolute value function, which is in the form of 

y = a |x-h| + k

the domain will be all real values.

For example, considering the graph given below.

Domain :

The horizontal spread of the graph, or the spreading over all the x-axis. So, domain is {-∞, ∞}.

Range :

The vertical spread of the graph, or the spreading is above 0 on y-axis. So, range is {0, ∞}.

From each graph :

(i)  Find vertex and write a function

(ii) State the domain and range

(iii) Intercept

(iv) axis of symmetry

Problem 1 :

Solution :

Equation of the function :

General form of absolute value function is y = a|x - h| + k

Vertex is at (-1, 0), the absolute value function is passing through the point (0, 2).

Applying the vertex,

y = a|x - (-1)| + 0

y = a|x + 1|

Applying the point (0, 2), we get

2 = a |0 + 1|

2 = a

So, the required function is

f(x) = 2|x + 1|

Domain :

All real value is domain.

Range :

[0, ∞), y ≥ 0

Axis of symmetry :

x = -1

Maximum / Minimum :

Minimum

Problem 2 :

Solution :

Equation of the function :

General form of absolute value function is y = a|x - h| + k

Vertex is at (3, 1), the absolute value function is passing through the point (4,0).

Applying the vertex,

y = a|x - 3| + 1

Applying the point (4, 0), we get

0 = a |4 - 3| + 1

-1 = a

a = -1

So, the required function is

f(x) = -|x - 3| + 1

Domain:

All real value is domain.

Range:

[1,-∞), y ≤ 1

Axis of symmetry:

x = 3

Maximum / Minimum

Maximum

Graph the function. Compare the graph to the graph of f(x) = ∣x∣. Describe the domain and range. 

Problem 3 :

f(x) = |x| - 4

Solution :

f(x) = |x| - 4

For any absolute value function, the domain is all real numbers. 

Comparing the given function with y = a|x - h| + k

(h, k) is (0, -4)

• a = 1
• Since it is positive, the graph will open up.
• Domain is (-∞, ∞)
• Range is [-4, ∞)

Problem 4 :

f(x) = |x + 1|

Solution :

f(x) = |x + 1|

Comparing the given function with y = a|x - h| + k

(h, k) is (-1, 0)

• a = 1
• Since it is positive, the graph will open up.
• Domain is (-∞, ∞)
• Range is [-1, ∞)

Problem 5 :

f(x) = (-3/2)|x|

Solution :

f(x) = (-3/2)|x|

Comparing the given function with y = a|x - h| + k

(h, k) is (0, 0)

• a = -3/2
• Since it is negative, the graph will open down.
• Domain is (-∞, ∞)
• Range is [0, -∞)

Problem 6 :

Describe and correct the error in graphing the function.

Solution :

Comparing the given function y = |x - 1| - 3 with y = a|x - h| + k 

(h, k) is (1, -3)

By observing the graph, the vertex of the absolute value function is (-1, -3). That 's the error.

Problem 6 :

Describe and correct the error in graphing the function.

Solution :

Comparing the given function y = -3|x| with y = a|x - h| + k 

(h, k) is (0, 0)

Since the sign of a is negative, the absolute value function must open down. By observing the graph, the function opens up. That is the error.

Problem 7 :

Compare the graphs. Find the value of h, k, or a.

Solution :

Comparing the graph in black, the blue graph is moved towards down of 3 units.

Describing the function g(x), we get

g(x) = |x| - 3

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

a = 1, h = 0 and k = -3

\Problem 8 :

Compare the graphs. Find the value of h, k, or a.

Solution :

Comparing the graph in black, the blue graph is moved towards right of 1 unit.

Describing the function g(x), we get

g(x) = |x - 1|

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

a = 1, h = 1 and k = 0