Are All Parabolas Similar?

Are All Parabolas Similar?

Student Outcomes

• Students apply the geometric transformation of dilation to show that all parabolas are similar.

New York State Common Core Math Algebra II, Module 1, Lesson 35

Worksheets for Algebra 2

Classwork

Exercises 1–8

1. Write equations for two parabolas that are congruent to the parabola given by 𝑦 = 𝑥², and explain how you determined your equations.
2. Sketch the graph of 𝑦 = 𝑥² and the two parabolas you created on the same coordinate axes.
3. Write the equation of two parabolas that are NOT congruent to 𝑦 = 𝑥². Explain how you determined your equations.

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4. Sketch the graph of 𝑦 = 𝑥² and the two non-congruent parabolas you created on the same coordinate axes.
5. What does it mean for two triangles to be similar? How do we use geometric transformations to determine if two triangles are similar?
6. What would it mean for two parabolas to be similar? How could we use geometric transformation to determine if two parabolas are similar?
7. Use your work in Exercises 1–6 to make a conjecture: Are all parabolas similar? Explain your reasoning.
8. The parabola at right is the graph of which equation?
   a. Label a point (𝑥, 𝑦) on the graph of 𝑃.
   b. What does the definition of a parabola tell us about the distance between the point (𝑥, 𝑦) and the directrix 𝐿, and the distance between the point (𝑥, 𝑦) and the focus 𝐹?
   c. Create an equation that relates these two distances.
   d. Solve this equation for 𝑥.
   e. Find two points on the parabola 𝑃, and show that they satisfy the equation found in part (d).

Discussion
Do you think that all parabolas are similar? Explain why you think so.
What could we do to show that two parabolas are similar? How might you show this?

Exercises 9–12
Use the graphs below to answer Exercises 9 and 10
9. Suppose the unnamed red graph on the left coordinate plane is the graph of a function 𝑔. Describe 𝑔 as a vertical scaling of the graph of 𝑦 = 𝑓(𝑥); that is, find a value of 𝑘 so that 𝑔(𝑥) = 𝑘𝑓(𝑥). What is the value of 𝑘? Explain how you determined your answer.
10. Suppose the unnamed red graph on the right coordinate plane is the graph of a function ℎ. Describe ℎ as a vertical scaling of the graph of 𝑦 = 𝑓(𝑥); that is, find a value of 𝑘 so that ℎ(𝑥) = 𝑘𝑓(𝑥). Explain how you determined your answer

Use the graphs below to answer Exercises 11–12
11. Suppose the unnamed function graphed in red on the left coordinate plane is 𝑔. Describe 𝑔 as a horizontal scaling of the graph of 𝑦 = 𝑓(𝑥). What is the value of the scale factor 𝑘? Explain how you determined your answer.
12. Suppose the unnamed function graphed in red on the right coordinate plane is h. Describe h as a horizontal scaling of the graph of 𝑦 = 𝑓(𝑥). What is the value of the scale factor 𝑘? Explain how you determined your answer.

Example: Dilation at the Origin
Let 𝑓(𝑥) = 𝑥2 and let 𝑘 = 2. Write a formula for the function 𝑔 that results from dilating 𝑓 at the origin by a factor of 1/2.
What would the results be for 𝑘 = 3, 4, or 5? What about 𝑘 = 1/2?

Lesson Summary

• We started with a geometric figure of a parabola defined by geometric requirements and recognized that it involved the graph of an equation we studied in algebra.
• We used algebra to prove that all parabolas with the same distance between the focus and directrix are congruent to each other, and in particular, they are congruent to a parabola with vertex at the origin, axis of symmetry along the 𝑦-axis, and equation of the form 𝑦 = 1/2𝑝 𝑥2.
• Noting that the equation for a parabola with axis of symmetry along the 𝑦-axis is of the form 𝑦 = 𝑓(𝑥) for a quadratic function 𝑓, we proved that all parabolas are similar using transformations of functions.

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