Parabola Graphs — SAT Math Explained
The U-shaped (or inverted U-shaped) graph of a quadratic function y = ax² + bx + c. The shape, direction, and position of the parabola encode all information about the quadratic.
The Core Idea
Every feature of the parabola has algebraic meaning. The vertex is the max/min; the axis of symmetry cuts the parabola in half; the x-intercepts are the roots; the y-intercept is c.
Key Vocabulary
The turning point — (h, k) — either the maximum or minimum of the function
The vertical line x = h passing through the vertex — the parabola is a mirror image across it
When a > 0 — vertex is the minimum
When a < 0 — vertex is the maximum
Where y = 0 — the roots; found by factoring or the quadratic formula
Where x = 0 — always equals c in y = ax² + bx + c
Finding Key Features
x = -b/(2a)
Substitute x = -b/(2a) back into the equation
x = -b/(2a) (same as vertex x-coordinate)
Larger |a| makes the parabola narrower; smaller |a| makes it wider
a > 0: opens up; a < 0: opens down
Vertex Form
y = a(x - h)² + k, where (h, k) is the vertex — vertex is immediately visible in this form
Real World Application
The height of a projectile as a function of time is a downward-opening parabola — vertex is peak height, x-intercepts are when it's at ground level
Common Errors to Avoid
Confusing which direction the parabola opens based on the sign of a
Forgetting the axis of symmetry is vertical (x = constant), not horizontal
Practice: Parabola Graphs
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The graph of y = x² opens in which direction?
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