A parabola is every point equally far from a fixed point, the focus, and a fixed line, the directrix.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Standard form
For (x − h)² = 4p(y − k) the vertex is (h, k) and the focus sits p units from it, inside the curve.
Why anyone cares
Every ray arriving parallel to the axis reflects through the focus. That is what a satellite dish and a headlight both exploit.
Hyperbolas
A hyperbola has two separate branches and comes from a difference of squared terms rather than a sum.
The asymptotes
For x²/a² − y²/b² = 1 the branches approach the lines y = ±(b/a)x but never reach them.
Telling them apart
One squared term means a parabola. Two added means a circle or an ellipse. Two subtracted means a hyperbola.
A parabola has one focus and a directrix
It is the set of points equidistant from a fixed point and a fixed line. That definition is why a parabolic dish focuses all incoming parallel rays to a single point — the reflection property follows from it.
A hyperbola is a difference of distances
Where an ellipse fixes the sum of distances to two foci, a hyperbola fixes the difference. That single change produces two separate branches opening away from each other.
Hyperbolas have asymptotes
Far from the centre a hyperbola approaches two straight lines. Sketching the asymptote box first and drawing the branches into its corners produces an accurate sketch quickly.
Telling them apart from the equation
Both squared terms positive and added gives a circle or ellipse; one subtracted gives a hyperbola; only one squared term gives a parabola. The signs of the squared terms classify the conic before any work.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Sana Read a Hyperbola
Sana is given x²/9 − y²/16 = 1.
- Step 1
The terms are subtracted, so this is a hyperbola.
Step 4: Your Turn
Practice makes it stick.
The Vertex
Problem 1 of 2
(x − 4)² = 8(y − 2). What is the x-coordinate of the vertex?
The Slope
Problem 2 of 2
x²/25 − y²/4 = 1. What is the positive asymptote slope, as a decimal?
Name That Conic
1 of 8
x² + y² = 25. Conic type? 1 circle, 2 ellipse, 3 parabola, 4 hyperbola.
2 of 8
x²/9 − y²/4 = 1. Conic type? 1, 2, 3 or 4.
3 of 8
y = x². Conic type? 1, 2, 3 or 4.
4 of 8
x²/16 + y²/9 = 1. Conic type? 1, 2, 3 or 4.
5 of 8
x²/4 − y²/9 = 1. What is a?
6 of 8
Same hyperbola. What is b?
7 of 8
Match each conic with what its equation looks like.
Tap a card on the left to start.
8 of 8
x²/36 − y²/36 = 1. What is the positive asymptote slope?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y²/16 − x²/9 = 1. What is a, the value under the leading positive term?
Question 2 of 2
What makes a parabola useful in a satellite dish?
What You Learned
- 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.