HOW TO WRITE AN EQUATION FOR A TRANSFORMATION WITH LINEAR EQUATIONS

Use the given functions to describe the listed transformations. Then write the equation of the transformed function. 

f(x) = 3x - 2

g(x) = -2x + 4

h(x) = 5

Problem 1 :

f(2x)

Solution :

Describing the transformation :

Horizontal shrink with the factor of 2 units.

f(x) = 3x - 2

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

= 6x - 2

Problem 2 :

2f(x)

Solution :

Describing the transformation :

f(x) is multiplied by 2, vertical stretch/shrink can happen. By observing the factor, 2 which is > 1. So, vertical stretch with the factor of 2.

New function :

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

= 6x - 4

Problem 3 :

f(x) - 2

Solution :

Describing the transformation :

f(x) is subtracted with 2, then vertical translation of 2 units down. 

New function :

f(x) = (3x - 2)

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

= 3x - 4

Problem 4 :

f(x + 2)

Solution :

Describing the transformation :

In the function x is added with 2

f(x + 2) = f(x - (-2))

Here h = -2

Move the curve left of 2 units.

f(x) = 3x - 2

Applying the transformation :

Moving the curve 2 units left

= 3(x - (-2)) - 2

= 3(x + 2) - 2

= 3x + 6 - 2

= 3x + 4

Problem 5 :

g(-x)

Solution :

Describing the transformation :

Since x is being changed as -x, there is a reflection across y axis.

g(x) = -2x + 4

g(-x) = -2(-x) + 4

= 2x + 4

Problem 6 :

(1/2) g(x)

Solution :

Describing the transformation :

The function is being multiplied with 1/2, then it must be vertical stretch or shrink. But based on the value we can check if it is stretch or shrink.

1/2 is lesser than 1, then it is vertical shrink.

g(x) = -2x + 4

(1/2) g(x) = (1/2) (-2x + 4)

= -x + 2

Problem 7 :

g(x) + 5

Solution :

Describing the transformation :

The function is added with 5, vertical translation of 5 units up.

Creating new function :

g(x) + 5 = -2x + 4 + 5

= -2x + 9

Problem 8 :

g(x - 5)

Solution :

Describing the transformation :

5 is subtracted with x, it should be the horizontal translation of 5 units to the right. Because in horizontal translation, we have to remember in opposite way.

Creating new function :

g(x - 5) = -2(x - 5) + 4

= -2x + 10 + 4

= -2x + 14

Problem 9 :

-3 g(x)

Solution :

Describing the transformation :

We have negative infront. So, it is reflection about x-axis. Vertical stretch with the factor of 3.

Creating new function :

g(x) = -2x + 4

-3 g(x) = -3(-2x + 4)

= 6x - 12

Problem 10 :

Using f, graph (a) g and (b) h. Describe the transformations from the graph of f to the graphs of g and h.

f(x) = 3x + 1; g(x) = f(x) − 2; h(x) = f(x − 2)

Solution :

g(x) = f(x) - 2

Describing the transformation :

The function is translating down 2 units.

Now we are going to get the new equation and check the same changes is happening when we draw the graph.

= 3x + 1 - 2

= 3x - 1

The new coordinates are (-2, -7) (-1, -4) (0, -1) and (1, 2)

h(x) = f(x − 2)

Describing transformation :

Moving the curve horizontally to the right of 2 units.

Creating the new function :

f(x) = 3x + 1

f(x - 2) = 3(x - 2) + 1

= 3x - 6 + 1

= 3x - 5

Problem 11 :

Graph f(x) = x and h(x) = (1/4)x − 2. Describe the transformations from the graph of f to the graph of h.

Solution :

Accordingly order of transformation, we have to do stretch or shrink, reflection and then translation.

Since x is multiplied by 1/4, and it is in between 0 to 1. It must be horizontal stretch. Vertical translation of 2 units down.