GRAPHING ABSOLUTE VALUE FUNCTIONS

About "Graphing absolute value functions"

"Graphing absolute value functions" is sometimes difficult job for the students who study high school math. 

Here we are going to see, "How to graph an absolute value function step by step"

Step 1 :

Before graphing any absolute value function, first we have to graph the parent function y = |x|

Its vertex is (0,0)

Let us take some random values for "x". 

x = -3 ----------------> y = |-3| = 3 -------------> (-3, 3)

x = -2 ----------------> y = |-2| = 2 -------------> (-3, 3)

x = -1 ----------------> y = |-1| = 1 -------------> (-3, 3)

x = 0  ----------------> y = |0| =  0  -------------> (0,0)

x = 1  ----------------> y = |1| =  1  -------------> (1, 1)

x = 2  ----------------> y = |2| =  2  -------------> (2, 2)

x = 3  ----------------> y = |3| =  3  -------------> (3, 3)

If we plot these points on the graph sheet, we will get a graph as given below.

When we look at the above graph,clearly the vertex is (0,0)

Step 2 :

Write the given absolute value function as

y - k = |x - h|

Step 3 :

Let x - h = 0 and y - k = 0.

Then we get x = h and y = k

Hence, the vertex is (h, k)

According to the vertex, we have to shift the above graph.

Additional notes :

If we have negative sign in front of absolute sign, we have to flip the curve over.

Example :   y = -|x|

Practice problems

Now, let us look at some practice problems on "Graphing absolute value function"

Problems 1 :

Graph the absolute value function given below.

y = |x-1|

Solution :

Then given function is in the form of y-k = |x-h|

To get vertex, let us equate y = 0 and x-1 = 0

Then, we get x =1 and  y = 0

Hence the vertex is (1, 0)

Hence, the graph of the given absolute value function is 

Let us loot at the next problem on "Graphing absolute value functions"

Problems 2 :

Graph the absolute value function given below.

y = |x-1| - 2

Solution :

Let us write the given absolute value function in the form

y - h = |x -h|

y = |x-1| - 2 -----------> y + 2 = |x-1|

To get vertex, let us equate y + 2 = 0 and x-1 = 0

Then, we get x = 1 and  y = -2

Hence the vertex is (1, -2)

Hence, the graph of the given absolute value function is 

Let us loot at the next problem on "Graphing absolute value functions"

Problems 3 :

Graph the absolute value function given below.

y = |x+3| + 3

Solution :

Let us write the given absolute value function in the form

y - h = |x -h|

y = |x+3| + 3 -----------> y - 3 = |x+3|

To get vertex, let us equate y - 3 = 0 and x + 3 = 0

Then, we get x = -3 and  y = 3

Hence the vertex is (-3, 3)

Hence, the graph of the given absolute value function is 

Let us loot at the next problem on "Graphing absolute value functions"

Problems 4 :

Graph the absolute value function given below.

y = |x-2|

Solution :

Then given function is in the form of y-k = |x-h|

To get vertex, let us equate y = 0 and x-2 = 0

Then, we get x =2 and  y = 0

Hence the vertex is (2, 0)

Hence, the graph of the given absolute value function is

Let us loot at the next problem on "Graphing absolute value functions"

Problems 5 :

Graph the absolute value function given below.

y = |x+4| + 3

Solution :

Let us write the given absolute value function in the form

y - h = |x -h|

y = |x+4| + 3 -----------> y - 3 = |x+4|

To get vertex, let us equate y - 3 = 0 and x + 4 = 0

Then, we get x = -4 and  y = 3

Hence the vertex is (-4, 3)

Hence, the graph of the given absolute value function is 

Let us loot at the next problem on "Graphing absolute value functions"

Problems 6 :

Graph the absolute value function given below.

y = |x-4| - 4

Solution :

Let us write the given absolute value function in the form

y - h = |x -h|

y = |x-4| -  4 -----------> y +4  = |x-4|

To get vertex, let us equate y + 4 = 0 and x - 4 = 0

Then, we get x = 4 and  y = -4

Hence the vertex is (4, -4)

Hence, the graph of the given absolute value function is 

Let us loot at the next problem on "Graphing absolute value functions"

Problems 7 :

Graph the absolute value function given below.

y = -|x-2| - 2

Solution :

Let us write the given absolute value function in the form

y - h = |x -h|

y = -|x-2| - 2 -----------> y + 2  = -|x-2|

To get vertex, let us equate y + 2 = 0 and x - 2 = 0

Then, we get x = 2 and  y = -2

Hence the vertex is (2, -2)

Important note : Since there is negative sign in front of the absolute sign, we have to flip the curve over. 

Hence, the graph of the given absolute value function is 

Let us loot at the next problem on "Graphing absolute value functions"

Problems 8 :

Graph the absolute value function given below.

y = -|x-4|

Solution :

Then given function is in the form of y-k = |x-h|

To get vertex, let us equate y = 0 and x - 4 = 0

Then, we get x = 4 and  y = 0

Hence the vertex is (4, 0)

Important note : Since there is negative sign in front of the absolute sign, we have to flip the curve over. 

Hence, the graph of the given absolute value function is 

Let us loot at the next problem on "Graphing absolute value functions"

Problems 9 :

Graph the absolute value function given below.

y = -|x| + 2

Solution :

Let us write the given absolute value function in the form

y - h = |x -h|

y = -|x| + 2 -----------> y - 2  = -|x|

To get vertex, let us equate y - 2 = 0 and x = 0

Then, we get x = 0 and  y = 2

Hence the vertex is (0, 2)

Important note : Since there is negative sign in front of the absolute sign, we have to flip the curve over. 

Hence, the graph of the given absolute value function is 

Let us loot at the next problem on "Graphing absolute value functions"

Problems 10 :

Graph the absolute value function given below.

y = -|x+1| + 3

Solution :

Let us write the given absolute value function in the form

y - h = |x -h|

y = -|x+1| + 3 -----------> y - 3  = -|x+1|

To get vertex, let us equate y - 3 = 0 and x+1 = 0

Then, we get x = -1 and  y = 3

Hence the vertex is (-1, 3)

Important note : Since there is negative sign in front of the absolute sign, we have to flip the curve over. 

Hence, the graph of the given absolute value function is 

Let us loot at the next problem on "Graphing absolute value functions"

Problems 11 :

Graph the absolute value function given below.

y = -|x| + 4

Solution :

Let us write the given absolute value function in the form

y - h = |x -h|

y = -|x| + 4 -----------> y - 4  = -|x|

To get vertex, let us equate y - 4 = 0 and x = 0

Then, we get x = 0 and  y = 4

Hence the vertex is (0, 4)

Important note : Since there is negative sign in front of the absolute sign, we have to flip the curve over. 

Hence, the graph of the given absolute value function is 

Let us loot at the next problem on "Graphing absolute value functions"

Problems 12 :

Graph the absolute value function given below.

y = -|x+1| -1

Solution :

Let us write the given absolute value function in the form

y - h = |x -h|

y = -|x+1| - 1 -----------> y + 1  = -|x+1|

To get vertex, let us equate y + 1 = 0 and x +1 = 0

Then, we get x = -1 and  y = -1

Hence the vertex is (-1, -1)

Important note : Since there is negative sign in front of the absolute sign, we have to flip the curve over. 

Hence, the graph of the given absolute value function is 

After having gone through the step by step solution for all the problems on "Graphing absolute value functions",  we hope that the students would have understood "How to graph absolute value functions".

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