Nonlinear Functions • Topic 1 of 3

Quadratic Functions

A quadratic function has the form f(x) = ax² + bx + c with a ≠ 0, and its graph is a parabola. Evaluating one means substituting the input and applying the exponent before multiplying — for f(x) = 2x² − 3x + 1, f(2) = 2(4) − 3(2) + 1 = 3. Watch signs when the input is negative, since squaring makes it positive. The sign of a tells you the parabola opens up (a > 0) or down (a < 0). Quadratics appear throughout SAT Advanced Math; here the focus is careful evaluation, using parentheses around the input to avoid sign and order-of-operations errors.

sat30t1 graphOvertex (0, 0)Quadratic functiony = x² — a parabola opening upward

✅ Solved examples

1. If f(x) = 2x² − 3x + 1, find f(2).
2(4) − 3(2) + 1 = 8 − 6 + 1 = 3.
2. If f(x) = x² + 4x, find f(3).
9 + 12 = 21.
3. If f(x) = x² − 5, find f(−2).
(−2)² − 5 = 4 − 5 = −1.
4. If f(x) = 3x² + 1, find f(0).
3(0) + 1 = 1.

✏️ Practice — try these, take hints as needed

1. If f(x) = x² + 2x, find f(4).
16 + 8.
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24.
2. If f(x) = 2x² − 1, find f(3).
2(9) − 1.
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17.
3. If f(x) = x² − 4x + 2, find f(2).
4 − 8 + 2.
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−2.
4. If f(x) = x² + 1, find f(−3).
(−3)² + 1.
9 + 1.
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10.
5. If f(x) = 3x² − 2x, find f(2).
3(4) − 2(2).
12 − 4.
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8.

📝 Topic test — 8 questions

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