Question 1: Function Transformation
The transformation between \( y = f(x) \) and \( y = f(-x) \) is known as a horizontal reflection or horizontal flip.
On a graph, this transformation reflects the graph of the function \( f(x) \) across the y-axis. If a function \( f(x) \) is symmetric about the y-axis, which means it is an even function (such that \( f(x) = f(-x) \)), then the graph will look the same after the transformation. If the function is not even, the shape will change and will be mirrored about the y-axis.
Question 2: Sketching the Parent Function and Its Reflection
For the parent function \( f(x) = x^2 \):
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Original function: The graph of \( f(x) = x^2 \) is a parabola opening upwards, with its vertex at the origin (0,0). The graph is symmetric about the y-axis.
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Reflected function: The graph of \( f(-x) = (-x)^2 = x^2 \) will look the same as \( f(x) \) because squaring a negative, as well as a positive number, yields the same result (since it's an even function).
Sketches:
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Sketch of \( y = f(x) = x^2 \):
\[ \begin{array}{c} \text{(0, 0)} \ \text{} \ \text{ } \ \text{ } \ \text{ } \ \text{ *} \ \text{(X-axis)} \ \end{array} \] (Parabola opens upward, vertex at (0,0), symmetric about the y-axis.)
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Sketch of \( y = f(-x) = x^2 \):
Since the function is \( y = x^2 \), the sketch will look exactly the same as the original function:
\[ \begin{array}{c} \text{(0, 0)} \ \text{} \ \text{ } \ \text{ } \ \text{ } \ \text{ *} \ \text{(X-axis)} \ \end{array} \] (It is also a parabola that opens upward and appears identical to the first.)
Labels:
- Label the first sketch as \( y = f(x) = x^2 \).
- Label the second sketch as \( y = f(-x) = x^2 \).
Both sketches will look identical because the parent function \( x^2 \) is an even function.
1 answer