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Question 2: 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 (. Create a fake graph for both the parent function and the transformation. Be sure to clearly label each sketch.
F(x)= 3√x
F (x) = 2^x
F (x) = x
f(x)= x^2
F (x) = √x

1 answer

Part A: Fill in the Blanks

  1. What is this type of transformation called?

    • Answer: This type of transformation is called a reflection.

  2. What does it look like on a graph?

    • Answer: Specifically, it is a reflection over the x-axis. On a graph, this means that every point on the graph of y = -f(x) is mirrored across the x-axis.

Part B: Sketch a Graph

Let's choose the parent function \( f(x) = x^2 \) for the graph illustrations.

Graph Sketches

Parent Function: \( f(x) = x^2 \)

  1. Description: The graph of \( f(x) = x^2 \) is a parabola that opens upward with its vertex at the origin (0,0).

    • Sketch:
          |
     4|                *
      |              *   *
     3|            *       *
      |          *           *
     2|        *               *
      |      *                   *
     1|    *                       *
      |  *                           *
  ---- +------------------------------ 
      -2 -1  0   1   2   3   4

Transformation: \( g(x) = -x^2 \)

  1. Description: The graph of \( g(x) = -x^2 \) is also a parabola, but it opens downward with its vertex at the origin (0,0).

    • Sketch:
          |
     4|  
      |                                   
     3|               
      |          
     2|             
      |               
     1|              
      |         *                       * 
  ---- +------------------------------
      -2 -1  0   1   2   3   4  
      |       *                   * 
     -1|         *               *
      |           *           *
     -2|             *       *
      |               *
     -3|                *
      |                  
     -4|

Labeling

  • Curve for \( f(x) = x^2 \): Label this curve as "Parent Function: \( f(x) = x^2 \)".
  • Curve for \( g(x) = -x^2 \): Label this curve as "Transformation: \( g(x) = -x^2 \)".

Conclusion

The sketches show the effect of reflecting the parent function \( f(x) = x^2 \) across the x-axis to create the function \( g(x) = -x^2 \).