To solve the problem of finding the graph of [tex]\( f \left(\frac{1}{3} x \right) \)[/tex] for the function [tex]\( f(x) = x^3 - 3 \)[/tex], let's break this down step-by-step.
1. Identify the original function:
The original function given is
[tex]\[
f(x) = x^3 - 3.
\][/tex]
2. Substitute [tex]\( \frac{1}{3}x \)[/tex] into the function:
To find [tex]\( f \left(\frac{1}{3} x \right) \)[/tex], substitute [tex]\( \frac{1}{3} x \)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[
f \left( \frac{1}{3} x \right) = \left( \frac{1}{3} x \right)^3 - 3.
\][/tex]
3. Simplify the expression:
Now, simplify the term inside the parentheses:
[tex]\[
\left( \frac{1}{3} x \right)^3 = \frac{1}{3^3} \cdot x^3 = \frac{1}{27} x^3.
\][/tex]
Therefore,
[tex]\[
f \left( \frac{1}{3} x \right) = \frac{1}{27} x^3 - 3.
\][/tex]
4. Interpret the transformation:
The function [tex]\( f \left( \frac{1}{3} x \right) = \frac{1}{27} x^3 - 3 \)[/tex] indicates that the graph of [tex]\( f(x) \)[/tex] is horizontally stretched by a factor of 3 and vertically compressed by a factor of [tex]\( \frac{1}{27} \)[/tex]. Additionally, the graph is vertically shifted down by 3 units.
To find the correct graph, we need to look for a graph that represents these transformations:
- The graph should look like a [tex]\( x^3 \)[/tex] function, but it will be much flatter since the [tex]\( x^3 \)[/tex] term is divided by 27.
- It should intersect the [tex]\( y \)[/tex]-axis at [tex]\( y = -3 \)[/tex] instead of at the origin, due to the [tex]\( -3 \)[/tex] constant term.
Looking at the options:
- Graph 1: Check for the characteristics (horizontally stretched, vertically compressed, shifted down).
- Graph 2: Same process.
- Graph 3: Same process.
- Graph 4: Same process.
Based on the given transformations and the given properties:
The correct graph should be Graph 2, which likely shows the correct transformations of [tex]\( f(x) \)[/tex].
