Question 1: Function Transformation Type
The transformation relating \( y = f(x) \) and \( y = f(-x) \) is called reflection across the y-axis.
On a graph, this transformation means that every point \((x, y)\) on the graph of \( y = f(x) \) will reflect to the point \((-x, y)\) on the graph of \( y = f(-x) \). As a result, the graph of \( f(-x) \) is a mirror image of the graph of \( f(x) \) with respect to the y-axis.
Question 2: Sketching Parent Functions
Let's choose the parent function \( f(x) = x^2 \) for this demonstration. The sketches for \( y = f(x) \) and \( y = f(-x) \) would look as follows:
1. Graph of \( y = f(x) = x^2 \)
The graph of \( y = x^2 \) is a parabola opening upwards.
- Key points to plot:
- Vertex at (0, 0).
- Points at (-2, 4), (-1, 1), (1, 1), and (2, 4).
2. Graph of \( y = f(-x) = (-x)^2 \)
The graph of \( y = f(-x) = x^2 \) is the same as \( y = x^2 \) because squaring \(-x\) also gives \( x^2 \). Therefore, the reflection across the y-axis does not change the graph.
- Key points to plot (same as above):
- Vertex at (0, 0).
- Points at (-2, 4), (-1, 1), (1, 1), and (2, 4).
Sketches
- Graph of \( y = x^2 \)
|
4 | *
| *
1 | *
| *
------|-----------------
-2 -1 0 1 2
- Graph of \( y = f(-x) = x^2 \)
|
4 | *
| *
1 | *
| *
------|-----------------
-2 -1 0 1 2
Summary
Both graphs are identical because \( f(x) = f(-x) \) for even functions, such as \( f(x) = x^2 \). This means that the function is symmetric about the y-axis.
1 answer