Parabolas are the graphs of quadratic functions, and the SAT tests them heavily. You need to move between different forms and pull out key information quickly.
The Three Forms of a Quadratic
| Form | Equation | What It Reveals |
|---|---|---|
| Standard | y-intercept () | |
| Vertex | Vertex () | |
| Factored | x-intercepts () |
In all forms, the sign of tells you the direction:
- : opens upward (minimum)
- : opens downward (maximum)
Finding the Vertex from Standard Form
For :
Then plug this back in to find .
Example 1: Find the vertex of
Vertex:
Reading Vertex Form
Example 2:
- Vertex: (watch the signs — )
- Opens downward ()
- Maximum value:
Finding x-Intercepts (Zeros)
Set and solve. You can use factoring, the quadratic formula, or completing the square.
Example 3: Find the zeros of
The Discriminant
The discriminant tells you how many x-intercepts:
- : two real zeros (parabola crosses x-axis twice)
- : one real zero (parabola touches x-axis)
- : no real zeros (parabola doesn't reach x-axis)
Example 4: How many x-intercepts does have?
No x-intercepts.
Axis of Symmetry
The axis of symmetry passes through the vertex:
Fun fact: the axis of symmetry is exactly halfway between the two x-intercepts (when they exist).
Example 5: If a parabola has zeros at and , the axis of symmetry is .
Practice Problems
Problem 1: What is the maximum value of ?
Solution
The vertex is and the parabola opens downward, so the maximum is .
Problem 2: Find the vertex of .
Solution
Vertex:
Problem 3: A ball is launched and its height is . What is the maximum height?
Solution
seconds
feet
Key Takeaways
- Know all three forms and what each one reveals
- Vertex from standard form: , then plug in to get
- Vertex form gives you the vertex directly:
- The sign of determines whether you have a max or min
- Use the discriminant to determine the number of x-intercepts
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