Parabola Shifts (Vertical Parabola)

Directions

1. Use the sliders to explore the parameters involved in determining the equation of the parabola.[br][br]2. Answer the questions below.

Question 1

Move point P around the parabola. What do you notice about the relationship between the point P (a point on the parabola) and the [b]Focus[/b] and [b]Directrix[/b]. [br]Note: The [b]Focus[/b] is a given point and the [b]Directrix[/b] is a given line. Together they define the parabola.

Question 2

What happens as you move the focus and directrix closer together? Farther apart?

Match My Equations

Click on the Match My Equation checkboxes and move the focus and directrix to match up the equations. You will notice that the equations are not written in the same form. Work with your partner to rewrite the equations so that you are convinced they are equivalent.[br]

Forms of Equations

You are already familiar with writing a quadratic function in the form [math]y=a\left(x-h\right)^2+k[/math]. The conics form is [br]The conics form of the parabola equation (the one you'll find in advanced or older texts) is: (credit to purplemath.com)[list]4[i]p[/i]([i]y[/i] – [i]k[/i]) = ([i]x[/i] – [i]h[/i])[sup]2[/sup] [br][/list]where p is the distance between the focus and the vertex.

Question 3

Use the 2 forms of writing the equation of a parabola (above) to determine the value of a (in terms of p - the distance of the vertex from the focus.) Use the "regular" form.

[math]a=4p[/math] [math]a=\frac{1}{4}p[/math] [math]a=\frac{1}{4p}[/math] [math]a=\frac{1}{4\sqrt{p}}[/math]

Question 4

Given the Focus (0,4) and the Directrix y = -2, what is the equation of the parabola? (Hint: Find the vertex first.)

[math]y=12x^2+1[/math] [math]y=\frac{1}{12}\left(x+1\right)^2[/math] [math]y=\frac{1}{12}x^2+1[/math] [math]\frac{1}{12}\left(y-1\right)=x^2[/math]

Use for Questions 5 - 7

An arch is being built in a park with a parabolic shape. The arch is to be 24 feet tall with a base width of 48 feet.

Question 5 (See information above.)

What is the equation of the parabolic arch? (Center the arch on the y-axis and let the x-axis represent the ground/base.)

[math]y=\frac{1}{24}\left(x+24\right)\left(x-24\right)[/math] [math]y=-\frac{1}{24}\left(x-24\right)\left(x+24\right)[/math] [math]y=-\left(x+24\right)\left(x-24\right)[/math] [math]y=\left(x+24\right)\left(x-24\right)[/math]

Question 6 (See information above.)

What are the focus and directrix of the arch?

Focus: (0,21); Directrix y = 27 Focus: (0,18); Directrix y = 30 Focus: (0,3); Directrix y = 6 Focus: (0,30); Directrix y = 18

Question 7 (See information above.)

What is the equation of the parabolic arch in "conics form"?

[math]-24\left(y-24\right)^2=x[/math] [math]-24\left(y-24\right)=x^2[/math] [math]-\frac{1}{24}y=x^2+24[/math] [math]-\frac{1}{24}\left(y-24\right)=x^2[/math]

Question 8

Given a parabola with a a focus of (2,5) and a directrix of y = -1, find the equation of the parabola.

Question 9 - BONUS

Given a parabola that contains the points (-1, 5) and (3, 8) and has the directrix y = -2, find the focus of the parabola. (Hint: Make your own Geogebra sheet to find the intersection of the equations you use to find the focus!)