Question 3: Consider the relationship between y=f(x) and y=f(-x).

Question

Part A fill in the blanks (3 points): What is this type of function transformation called? What does it look like on a graph?
Answer: This type of transformation is called a [blank]. Specifically, it is a reflection over the [blank]-axis. On a graph, this means that every point on the graph of y=f(-x) [blank].
Part B Sketch a graph (7 points):
Choose one of the parent functions from question 1.(F(x)= 3√x
F (x) = 2^x
F (x) = x
f(x)= x^2
F (x) = √x

)create a full visual fake graph. both the parent function and the transformation. Be sure to clearly label each sketch.

Answer 1

Part A

Answer: This type of transformation is called a horizontal reflection. Specifically, it is a reflection over the y-axis. On a graph, this means that every point on the graph of \( y = f(-x) \) is mirrored across the y-axis compared to the graph of \( y = f(x) \).

Part B - Sketch

Let's choose the parent function \( f(x) = x^2 \) and reflect it to create the transformation \( f(-x) = (-x)^2 = x^2 \).

1. Graph of Parent Function \( f(x) = x^2 \)

• The parabola opens upwards.
• It has its vertex at the origin (0,0).
• The points on the graph include:
  • (-2, 4)
  • (-1, 1)
  • (0, 0)
  • (1, 1)
  • (2, 4)

2. Graph of Transformation \( f(-x) = x^2 \)

• The reflection of \( f(x) = x^2 \) over the y-axis will result in the same graph.
• Thus, the transformation \( f(-x) = x^2 \) also opens upwards and has the same vertex at the origin (0,0).
• The points will also include:
  • (-2, 4)
  • (-1, 1)
  • (0, 0)
  • (1, 1)
  • (2, 4)

Because the equation for \( f(x) \) and \( f(-x) \) are identical in this case, both graphs will look the same.

Here’s a simple sketch:

y
↑
|                *                      *
|              *   *                  *   *
|            *       *              *       *
|          *           *          *           *
|        *               *      *               *
|      *                   *  *                   *
|    *                       *                     *
+------------------------------------------------→ x
    -2   -1   0    1    2

Labels

• The curve above is labeled as f(x) = x².
• Since f(-x) is the same as f(x) for this function, we don’t need a separate curve.

For any other selected parent function: The process to create the graph and determine its points would be similar, showing the reflection property clearly. Each graph for the transformed function will display the original function mirrored across the y-axis, if the original function is not even.

Feel free to select a different parent function from your list if that helps showcase the reflection better!