Parabolas and Hyperbolas

The other two conic sections.

12 problems · about 22 minutes · TEKS P.3.B, TEKS P.3.E

What this lesson teaches

The student identifies the vertex, focus, centre and asymptotes of parabolas and hyperbolas from their equations.

One squared term gives a parabola; two added give a circle or ellipse; two subtracted give a hyperbola.
A parabola reflects every parallel ray through its focus.
A hyperbola approaches y = ±(b/a)x without ever arriving.

Straightforward practice. Get the method working first.

5 problems

y²/16 − x²/9 = 1. What is a, the value under the leading positive term?
Answer 4
Why 4.
What makes a parabola useful in a satellite dish?
Answer Parallel rays all reflect through the focus.
Why Every parallel ray reflects through the single focus.
x² + y² = 25. Conic type? 1 circle, 2 ellipse, 3 parabola, 4 hyperbola.
Answer 1
Why Added squares with equal denominators.
x²/9 − y²/4 = 1. Conic type? 1, 2, 3 or 4.
Answer 4
Why Subtracted squares.
y = x². Conic type? 1, 2, 3 or 4.
Answer 3
Why Only one variable is squared.

The same ideas with more to keep track of.

3 problems

x²/16 + y²/9 = 1. Conic type? 1, 2, 3 or 4.
Answer 2
Why Added squares with different denominators.
x²/4 − y²/9 = 1. What is a?
Answer 2
Why √4.
Same hyperbola. What is b?
Answer 3
Why √9.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each conic with what its equation looks like.
Answer Parabola → Only one variable squared; Ellipse → Two squares added; Hyperbola → Two squares subtracted
Why Count the squared terms and check the sign between them.
x²/36 − y²/36 = 1. What is the positive asymptote slope?
Answer 1
Why b ÷ a with equal values.
The Vertex: (x − 4)² = 8(y − 2). What is the x-coordinate of the vertex?
Answer 4
Why 4.
The Slope: x²/25 − y²/4 = 1. What is the positive asymptote slope, as a decimal?
Answer 0.4
Why 0.4.