The following diagram shows the vertex form of a parabola. Scroll down the page for more examples and explanations about the vertex form of the equation of a parabola.
Are all parabolas congruent? Use the following questions to support your answer.
a. Draw the parabola for each focus and directrix given below.
b. What do we mean by congruent parabolas?
c. Are the two parabolas from part (a) congruent? Explain how you know.
d. Are all parabolas congruent?
e. Under what conditions might two parabolas be congruent? Explain your reasoning
Example 1: Derive an Equation for a Parabola
Consider a parabola 𝑃 with distance 𝑝 > 0 between the focus with coordinates (0,1/2 𝑝), and directrix 𝑦 = −1/2 𝑝.
What is the equation that represents this parabola?
THEOREM: Given a parabola 𝑃 given by a directrix 𝐿 and a focus 𝐹 in the Cartesian plane, then 𝑃 is congruent to the graph of 𝑦 = 1/2𝑝 𝑥2, where 𝑝 is the distance from 𝐹 to 𝐿.
Exercises 6–9: Reflecting on the Theorem
6. Restate the results of the theorem from Example 2 in your own words.
7. Create the equation for a parabola that is congruent to 𝑦 = 2𝑥2. Explain how you determined your answer.
8. Create an equation for a parabola that IS NOT congruent to 𝑦 = 2𝑥2. Explain how you determined your answer.
9. Write the equation for two different parabolas that are congruent to the parabola with focus point (0,3) and directrix line 𝑦 = −3.
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