WRITE THE LINEAR FUNCTION FOR THE GIVEN TRANSFORMATION OF THE FUNCTION

Write a function g in terms of f so that the statement is true.

Problem 1 :

The graph of g is a horizontal translation 2 units right of the graph of f.

Solution :

Let g(x) and f(x) will be the given functions.

Transformation done :

Horizontal translation of 2 units right. Since it is horizontal, we have to remember in opposite way. So, the required numerical value we will use -2.

g(x) = f(x - 2)

So, the required function g in terms of f is f(x - 2).

Problem 2 :

The graph of g is a reflection in the y-axis of the graph of f.

Solution :

Transformation done :

Reflection in the y-axis. Since reflection across y-axis, change x as -x.

g(x) = f(-x)

Problem 3 :

The graph of g is a vertical stretch by a factor of 4 of the graph of f.

Solution :

Transformation done :

Vertical stretch by a factor of 4

g(x) = 4 f (x)

Problem 4 :

The graph of g is a horizontal shrink by a factor of 1/5 of the graph of f

Solution :

Transformation done :

Horizontal shrink by a factor of 1/5. Since it is horizontal transformation, we have to use opposite numerical value.

g(x) = f (5x)

Problem 5 :

The graph of h(x) = a ⋅f(bx − c) + d is a transformation of the graph of the linear function f. Select the word or value that makes each statement true.

a) The graph of h is a ______ shrink of the graph of f when a = 1/3 , b = 1, c = 0, and d = 0.

b) The graph of h is a reflection in the ______ of the graph of f when a = 1, b = −1, c = 0, and d = 0.

c) The graph of h is a horizontal stretch of the graph of f by a factor of 5 when a = 1, b = _____, c = 0, and d = 0.

Solution :

a) Here a = 1/3 , b = 1, c = 0, and d = 0. Applying all values in the function h(x), we get

h(x) = (1/3) ⋅ f(1x − 0) + 0

g(x) = (1/3) ⋅ f(x)

The value of a will say the vertical stretch or shrink. Since a lies in between 0 and 1. So, the answer is vertical shrink.

b)  when a = 1, b = −1, c = 0, and d = 0.

h(x) = 1 ⋅ f(-x − 0) + 0

h(x) = f(-x)

Since x is changed as -x, it must be reflection across y-axis. 

c) h is a horizontal stretch of the graph of f by a factor of 5 when a = 1, b = _____, c = 0, and d = 0

Since it is horizontal transformation, we have to use the numbers in opposite way. The given factor is 5, but we have to use the value of b as 1/5.

Problem 6 :

The graph of g(x) = a ⋅f(x − b) + c is a transformation of the graph of the linear function f. Select the word or value that makes each statement true.

a. The graph of g is a vertical ______ of the graph of f when a = 4, b = 0, and c = 0.

b. The graph of g is a horizontal translation ______ of the graph of f when a = 1, b = 2, and c = 0.

c. The graph of g is a vertical translation 1 unit up of the graph of f when a = 1, b = 0, and c = ____.

Solution :

a) When a = 4, b = 0, and c = 0. Applying all the values in the function g(x), we get

g(x) = a ⋅f(x − b) + c

g(x) = 4 ⋅f(x − 0) + 0

 = 4 ⋅f(x)

Here 4 is greater than 1 and vertical transformation. So, we can use the word stretch.

The answer is vertical stretch

b)  when a = 1, b = 2, and c = 0.

g(x) = 1 ⋅f(x − 2) + 0

g(x) = f(x − 2)

Horizontal translation of 2 units to the right.

c)  a = 1, b = 0, and c = ____.

g(x) = 1 ⋅f(x − 0) + c

Vertical translation of 1 unit, since it is positive 1 we understand that we move up. So, the answer is vertical translation of 1 unit up.