To understand how the graph of \( f(x) = x^2 \) transforms into \( y = f(3x) \), we first recognize that compressing a graph horizontally involves changing the input to the function. Specifically, when you replace \( x \) with \( 3x \), it compresses the function by a factor of \( \frac{1}{3} \).
The parent function \( f(x) = x^2 \) produces points that can be calculated as follows:
- \( f(-1) = (-1)^2 = 1 \)
- \( f(0) = 0^2 = 0 \)
- \( f(1) = 1^2 = 1 \)
The graph of \( f(x) = x^2 \) gives us the following points:
- (-1, 1)
- (0, 0)
- (1, 1)
When we transform it to \( y = f(3x) = (3x)^2 = 9x^2 \), we need to identify how this affects specific \( x \) values.
To find the new \( y \) outputs at the same \( x \) values:
- For \( x = -\frac{1}{3} \), \( f(3(-\frac{1}{3})) = f(-1) = 1 \) ⇒ point is (-1/3, 1)
- For \( x = 0 \), \( f(3(0)) = f(0) = 0 \) ⇒ point is (0, 0)
- For \( x = \frac{1}{3} \), \( f(3(\frac{1}{3})) = f(1) = 1 \) ⇒ point is (1/3, 1)
As we see, the \( y \)-values for \( x = -\frac{1}{3} \) and \( x = \frac{1}{3} \), both yield \( y = 1 \).
Now, let's look at the provided graphs to identify the one that has characteristics compatible with these transformations:
- Graph #1 has dots at (-1, 9) and (1, 9). These points are too high for the transformed function.
- Graph #2 has dots at (-3, 1) and (1, 3). These don't align with the computed points.
- Graph #3 has dots at (-1, 4) and (1, 4). Again, not compatible.
- Graph #4 has dots at (-1, 1) and (1, 1). This matches our transformed points when evaluating \( f(3x) \) for \( x = -\frac{1}{3} \) and \( x = \frac{1}{3} \).
The answer is Graph #4 which correctly reflects the horizontal compression resulting in a similar shape to the parent function \( f(x) = x^2 \), maintaining the correct height at \( y = 1 \) for these new \( x \) values.