Question

What is the graph of g?

Answer 1

Vertical Compression by factor of 1/4

Explanation:

Two methods:

Method 1. Transformations

Method 2. Algebraic input-output tables

Method 1. Transformations

The Main concept of this question is about Transformations of functions -- specifically, multiplying on the outside by a positive number less than 1.

The transformation that occurs when multiplying a function by a positive number on the outside of the function is a vertical stretch or compression.

Positive numbers larger than 1 will stretch it vertically, whereas positive numbers smaller than 1 will compress it vertically.

Therefore, multiplying by 1/4 on the outside, a positive number less than 1, will vertically compress the function down to one-fourth the size.

This means that for g(x), all points on the original function f will have their heights reduced to 1/4 their original height (or depth) -- making all points on g(x) 1/4 their previous distance from the x-axis on the "f" function.

Method 2. Algebraic input-output tables

Observe on the graph three points on the function f:

• (0,0), (1,4) and (3,0) --- points on the function "f"
• In function notation, this means
  `f(0)=0`,
  `f(1)=4`, and
  `f(3)=0`

Using the equation relating f and g,
`g(x)=(1)/(4)f(x)`, we can find how those points would look like on the new function g(x).

For
`f(0)=0`

`g(0)=(1)/(4)[f(0)]\\g(0)=(1)/(4)[0]\\g(0)=0`

For
`f(1)=4`

`g(1)=(1)/(4)[f(1)]\\g(1)=(1)/(4)[4]\\g(1)=1`

For
`f(3)=0`

`g(3)=(1)/(4)[f(3)]\\g(3)=(1)/(4)[0]\\g(3)=0`

These known points should correctly identify the graph from the possible choices.