Quadratic Functions • Topic 1 of 5

Parabolas

The graph of a quadratic function y = ax² + bx + c is a parabola, a symmetric U-shaped curve. The sign of a sets the direction: a > 0 opens upward (a lowest point), a < 0 opens downward (a highest point). The larger |a| is, the narrower the parabola. Every parabola has a single turning point, the vertex, and a vertical axis of symmetry through it. The y-intercept is (0, c). Recognising these features from the equation lets you sketch and read parabolas quickly, which the SAT tests in both algebra and graph-interpretation questions.

✅ Solved examples

1. Does y = 2x² − 3 open up or down?
a = 2 > 0, so it opens upward.
2. Does y = −x² + 4x open up or down?
a = −1 < 0, so it opens downward.
3. What is the y-intercept of y = x² + 2x + 5?
At x = 0, y = 5, so (0, 5).
4. Which is narrower, y = x² or y = 3x²?
Larger |a| is narrower, so y = 3x² is narrower.

✏️ Practice — try these, take hints as needed

1. Does y = −2x² + 1 open up or down?
Check the sign of a.
a = −2 < 0.
—
Downward.
2. Does y = 5x² − 2x open up or down?
Sign of a.
a = 5 > 0.
—
Upward.
3. What is the y-intercept of y = 2x² − 7?
Set x = 0.
y = −7.
—
(0, −7).
4. Which is wider, y = x² or y = (1/2)x²?
Smaller |a| is wider.
1/2 < 1.
—
y = (1/2)x².
5. How many turning points does a parabola have?
It is a single U-shape.
One vertex.
—
One.

📝 Topic test — 8 questions

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