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

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

1 answer

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 points on the graph of y = f(x).

Part B

Let's choose the function \( f(x) = x^2 \) as our parent function.

Graph Sketch:

To create the graphs, we will sketch both \( f(x) = x^2 \) and its transformation \( f(-x) = (-x)^2 = x^2 \).

  1. Parent Function:

    • The graph of \( y = x^2 \) is a parabola that opens upwards with its vertex at the origin (0,0).
    • Points to plot: (0,0), (1,1), (-1,1), (2,4), (-2,4).

  2. Transformation:

    • The graph of \( y = f(-x) = x^2 \) will look identical to the parent function because squaring a negative number gives the same result as squaring a positive number.
    • Points to plot: The same points as the parent function, confirming the reflection: (0,0), (1,1), (-1,1), (2,4), (-2,4).

Sketch Representation:

Since I cannot draw here, I can describe how you can sketch this on graph paper.

  • For both functions, plot the points mentioned above.
  • Draw a smooth curve through these points to create the parabola for \( y = x^2 \).
  • Clearly label the first graph as "y = x^2" (Parent Function).
  • For the second graph, do the same with the same points, and label it again as "y = f(-x)".

Remember:

  • The reflection over the y-axis means that both graphs will look identical.
  • Use a different color for the transformation if desired to highlight that they are the same function.