Algebra 2
4.2 Notes
4.2 Parabolas
Date: __________
Exploring Parabolas
Parabola: a set of points equidistant from a line, the directrix, and a point, the focus.
The Focus always lies on the axis of _________________________________ .
The Directrix will always be __________________________________ to the axis of symmetry.
Writing the Equation of a Parabola with Vertex at (0, 0)
Learning Target E: I can write the equation for and graph a parabola with its vertex at (0, 0).
Vertical
Horizontal
Parabolas with Vertices at the Origin, (0, 0)
Vertical
Horizontal
Standard Form Equation:
Standard Form Equation:
If p > 0, opens ___________________________
If p > 0, opens ___________________________
If p < 0, opens ___________________________
If p < 0, opens ___________________________
Focus:
Focus:
Directrix:
Axis of Symmetry:
Directrix:
Axis of Symmetry:
Reflect. Why is the directrix placed at −𝑝?
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Algebra 2
4.2 Notes
Find the equation of the parabola from the description of the focus and directrix. Then make
a sketch showing the parabola, the focus, and the directrix.
A. Focus (-8, 0), directrix x = 8
Step 1: Determine whether the parabola is vertical or horizontal.
Step 2: Confirm the vertex is at (0, 0).
Is the y-coordinate of the vertex the same as the focus?
Is the x-coordinate halfway between the focus and the directrix?
Step 3: Write and simplify the equation for the parabola.
Step 4: Plot the focus and directrix and sketch the parabola.
B. Focus (0, -2), d: y = 2.
C. Focus (2, 0), d: x = -2.
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D. Focus (0, − 2), d: 𝑦 = 2
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Algebra 2
4.2 Notes
Writing the Equation of a Parabola with Vertex at (h, k)
Learning Target F: I can write an equation for and graph a parabola with vertex at (h, k).
The standard form equation for a parabola with a vertex (h, k) can be found by translating from
(0, 0) to (h, k): substitute (x - h) for x and (y – k) for y. This also translates the focus and directrix
each by the same amount.
Parabolas with Vertex at (h, k)
Vertical
Horizontal
Standard Form Equation:
Standard Form Equation:
p=
p=
Focus:
Directrix:
Axis of Symmetry:
Focus:
Directrix:
Axis of Symmetry:
Find the equation of the parabola from the description of the focus and directrix. Then make a
sketch showing the parabola, focus, and the directrix.
A. Focus (3, 2), directrix y = 0.
B. Focus (-1, -1), directrix x = 5.
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Algebra 2
4.2 Notes
C. Focus (5, -1), directrix 𝑥 = −3
D. Focus (-2, 0), directrix 𝑦 = 4
Rewriting the Equation of a Parabola to Graph the Parabola
Learning Target G: I can convert a general form equation of a parabola to standard form by
completing the square and graph the parabola, focus and directrix.
Just like we needed to rewrite the equation of a circle in standard form in order to graph it, we need
to make sure the equation of a parabola is in standard form. We will utilize completing the square
to do this.
Convert the equation to the standard form of a parabola. Then graph the parabola, the focus,
and the directrix.
A. 𝑥 2 − 4𝑥 − 4𝑦 + 12 = 0
B. 𝑦 2 + 2𝑥 + 8𝑦 + 18 = 0
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Algebra 2
4.2 Notes
Solving a Real-World Problem
Learning Target H: I can use a parabola to solve a real-world problem.
Parabolic shapes occur in a variety of applications in science and engineering that take
advantage of the concentrating property of reflections from the parabolic surface at the
focus.
A. Parabolic microphones are so-named because they use a parabolic dish to bounce sound waves
toward a microphone placed at the focus of the parabola in order to increase sensitivity. The dish
shown has a cross section dictated by the equation 𝑦 2 = 32𝑥 where x and y are in inches. How from
from the center of the dish should be microphone be placed?
B. A reflective telescope uses a parabolic mirror to focus light rays before creating an image with
the eyepiece. If the focal length (the distance from the bottom of the mirror’s bowl to focus) is 140
mm and the mirror has a 70 mm diameter (width), what is the depth of the bowl of the mirror?
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