REFLECTION OF ABSOLUTE VALUE FUNCTION FOR A GIVEN FUNCTION

Problem 1 :

Let

f(x) = ∣ x + 3 ∣ + 1.

a) Write a function g whose graph is a reflection in the x-axis of the graph of f.

b) Write a function h whose graph is a reflection in the y-axis of the graph of f.

Solution :

f(x) = ∣ x + 3 ∣ + 1.

Finding reflection on x-axis :

Let g(x) be the reflected function about x-axis.

Put g(x) = -f(x)

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

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

So, the reflected function is g(x) = -|x + 3| - 1.

Finding reflection on y-axis :

Let h(x) be the reflected function about y-axis.

Put h(x) = f(-x)

Put x = -x

h(x) = ∣-x + 3 ∣ + 1

Factoring negative from absolute function.

h(x) = ∣-(x - 3) ∣ + 1

h(x) = ∣x - 3∣ + 1

So, the reflected function is h(x) = ∣x - 3∣ + 1

Problem 2 :

Find reflection of the function f(x) about x-axis.

f(x) = − ∣x + 2∣ − 1

Solution :

f(x) = − ∣x + 2∣ − 1

Let g(x) be the reflected function.

g(x) = -f(x)

g(x) = − (− ∣x + 2∣ − 1)

g(x) = ∣x + 2∣ + 1

So, the reflected function g(x) is ∣x + 2∣ + 1.

Problem 3 :

f(x) = ∣6x∣ − 2; reflection in the y-axis

Solution :

Let g(x) be the reflected function about y-axis.

g(x) = f(-x)

g(x) = ∣6(-x)∣ − 2

g(x) = ∣-6x∣ − 2

g(x) = ∣6x∣ − 2

So, the reflected function g(x) is ∣6x∣ − 2.

Problem 4 :

f(x) = ∣2x − 1∣ + 3; reflection in the y-axis

Solution :

Let g(x) be the reflected function about y-axis.

g(x) = f(-x)

g(x) = ∣2(-x) - 1∣ + 3

g(x) = ∣-2x - 1∣ + 3

g(x) = ∣-(2x + 1)∣ + 3

g(x) = ∣2x + 1∣ + 3

So, the reflected function g(x) is ∣2x + 1∣ + 3

Problem 5 :

f(x) = −3 + ∣x − 11∣ ; reflection in the y-axis

Solution :

Let g(x) be the reflected function about y-axis.

g(x) = f(-x)

g(x) = −3 + ∣x − 11∣

g(x) = −3 + ∣-x − 11∣

g(x) = -3 + |-(x + 11)|

g(x) = -3 + |x + 11|

So, the reflected function g(x) is -3 + |x + 11|.