Transformations of Parent Functions 

You know that functions can be used to model many real-world situations. Transformations allow functions to be more flexible in the situations that they model.

Use the distances shown in the table to sketch your project. Identify points on the coordinate plane and use these points to find a function that represents your structure. Finally, confirm your function using the graphing tool.

Distance from Toll Booth/Control Toom to Center of Structure: 
Radio Telescope: 100 m
Arched Bridge: 100 m

Height of Support Structure:
Radio Telescope: 50 m
Arched Bridge: N/A

Maximum Height/Depth:
Radio Telescope: 10 m
Arched Bridge: 20 m

Maximum Width:
Radio Telescope: 60 m
Arched Bridge: 60 m

Your choice: (1 point)

_____ Radio Telescope Dish _____ Arched Bridge

Your sketch:

1. Make a sketch of your project. This will be either an upward or downward turned parabola. Show the measurements from the table on your sketch. (3 points)

2. Sketch your project on a coordinate plane. The toll booth (bridge project) and control room (dish project) should be located at the origin.

2a. First identify the vertex of the parabola. This will be at the center of the bridge or dish. Label this point with its coordinates. (1 point)

2b. Locate the endpoints of your structure. Use the width of the structure to determine where on the parabola it starts and stops. Label these points with their coordinates. (2 points)

2c. Sketch a parabola through the labeled points. Highlight the part of the parabola that represents your structure. (1 point)

Your function:

3. What is the parent function for your graph? (1 point)

4. The parent function will be stretched and shifted horizontally and vertically. Transform the parent function to represent your structure. (3 points; 1 point each part)

4a. The parent function includes a stretch or compression, a. Multiply the parent function by a. (You will find the value of a later.)

4b. To move the graph up or down, use the transformation f(x) + k. Write the function that has the same vertical translation as your project. HINT: Use the y-coordinate of the vertex to identify the vertical shift.

4c. To move the graph to the right or left, use the transformation f(x – h). Change the function you wrote in part 4b to have the same horizontal translation as your project. HINT: Use the x-coordinate of the vertex to identify the horizontal shift.

5. Complete your function by solving for a. To do this, substitute the coordinates of one of the endpoints that you identified in part 2b into your transformed function.

5a. What is a? (2 points)

5b. Use this value of a to write the function that represents your structure. (1 point) Rewrite your function in standard form. HINT: First square the binomial, then multiply each term by the constant, a. Round the coefficients and constant to 2 decimal places. (2 points)

Confirm your function:

6. Use the Graphing Tool to graph your function. First click the parabola symbol, and then two points on the graph. For a parabola that opens upward, first select the vertex and then the right endpoint. For a downward opening parabola, select the vertex and then the left endpoint.

When you are satisfied that the graph matches your sketch, write down the function displayed on the graphing tool. (2 points)

7. Compare your function to the Graphing Tool function. Are the signs the same? Are the coefficients reasonably close? (1 point)