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 graph with all positive, negative, positive-negative, negative-positive areas on the graph. both the parent function and the transformation. Be sure to clearly label each sketch.

Answer 1

Part A

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 \( y = f(-x) \) has its x-coordinates negated, resulting in a mirror image of the graph of \( y = f(x) \) across the y-axis.

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Part B

Let's choose one of the parent functions to sketch. We'll take \( f(x) = x^2 \) (the quadratic function) for this example.

1. Sketch the Parent Function \( y = x^2 \):

   • The graph is a parabola that opens upwards with its vertex at the origin (0,0).
   • It takes values such that \( y \geq 0 \) for all \( x \).
   • The points include:
     • (−2, 4)
     • (−1, 1)
     • (0, 0)
     • (1, 1)
     • (2, 4)

2. Sketch the Transformed Function \( y = f(-x) = (-x)^2 \):

   • Since squaring a number results in the same output regardless of the sign of the input, \( y = f(-x) = (-x)^2 = x^2 \).
   • Therefore, the graph of \( y = f(-x) \) looks identical to \( y = f(x) \), and it also opens upwards with a vertex at the origin (0,0).
   • The points will remain the same:
     • (-2, 4)
     • (-1, 1)
     • (0, 0)
     • (1, 1)
     • (2, 4)

Graph

Here's how you could sketch these graphs:

1. Draw the Cartesian coordinates on the graph (x-axis and y-axis).
2. Plot the points for both \( y = x^2 \) and \( y = f(-x) \), noting that they overlap.
3. Label both curves clearly:
   • Label the original function in blue: \( y = x^2 \)
   • Label the transformed function in red: \( y = f(-x) = x^2 \)

Example of what it would look like:

   |        *         *        
   |      *   *     *   *      
   |    *       * *       *    
   |  *                     *   
---+-------------------------   
   |*                        *   
   |                          * 
   |                          * 
   |___________________________
   -3  -2  -1  0   1   2   3 

In this sketch:

• The asterisks represent points on the graph of \( y = x^2 \) (and thus the same points for \( y = f(-x)\)).
• Each curve simply mirrors perfectly over the y-axis, showing that \( f(-x) \) results in no change for the quadratic function.

Make sure to name the axes and provide a key for the colors if coloring the lines for better visualization!