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. both the parent function and the transformation. Be sure to clearly label each sketch.

Answer 1

Part A: Fill in the blanks

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 relative to the graph of y = f(x).

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Part B: Sketch a graph

Let's choose the parent function \( f(x) = x^2 \) (the quadratic function) for this part.

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

   • The parent function is a parabola opening upwards with its vertex at the origin (0, 0).
   • Points to plot:
     • (0, 0)
     • (1, 1)
     • (2, 4)
     • (-1, 1)
     • (-2, 4)

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

   • The transformation \( f(-x) \) results in the exact same graph as the parent function because the parabola is symmetric about the y-axis.

Sketch Description:

• Draw a coordinate plane with the x-axis and y-axis labeled.
• Plot the points for \( f(x) = x^2 \).
• Since the graph of \( f(-x) = x^2 \) is the same as \( f(x) \), it will overlap exactly.
• Label the graph of \( f(x) = x^2 \) as the "Parent Function" and the graph of \( f(-x) \) also as "Transformed Function".

Final Note: Both functions will look the same in this case, as the quadratic function is symmetric about the y-axis.