You know that functions can be used to model any number of phenomena and systems. Transformations allow functions to be more flexible in the situations that they model. Use the Graphing Tool to explore the function for your parabola.
1. Sketch a picture of your project on a coordinate plane, including measurements. (3 points)
2. Now sketch a graph of the parabola. Without doing any calculations, you should be able to label the axes and three points. (3 points)
The Parent Function
3. Before performing any transformations, write the equation for the parent function of your parabola as a function f(x). (1 point)
4. Use the Graphing Tool to draw the parent function. First click the parabola symbol, and then click the vertex and another point on the graph. Identify the points you chose. HINT: Use an x-value of 10 for your second point. (1 point)
Your Transformations: Moving the Vertex
5. To move the equation to the right or left, you can use the transformation f(x − h). Change the parent function to have the same horizontal translation as your project. (2 points)
6. Confirm that your transformation is accurate: Use the Graphing Tool and apply the same horizontal shift to the parent function. Select a point on the graph and plug the values into your equation. Show your work. (2 points: 1 point for an ordered pair, 1 point for an accurate equation)
7. To move the equation up or down, you can use the transformation f(x) + k. Write the equation that you would use to give your new function the same vertical translation as your project. (2 points)
8. Confirm that your transformation is accurate: Use the Graphing Tool and apply the same vertical shift to the parent function. Select a point on the graph and plug the values into your equation. Show your work. (2 points: 1 point for an order pair, 1 point for an accurate equation)
Your Transformations: Stretching the Graph
So far you have translated your parent function horizontally and vertically to position the vertex of your project. How much should you stretch your parent function for your equation to fit the project?
9. In the equation , which variable is responsible for stretching, compressing, or reflecting the shape of the graph? (1 point)
10. Using the equation from question 8, plug in a coordinate pair from question 2 to solve for the unknown stretching/compressing/reflecting variable. Do not choose the vertex. (2 points)
11. What is the complete equation for your parabola? (1 point)