Math

Nonlinear Functions

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Nonlinear functions show up all over the Math section — parabolas, exponentials, and curves that bend. Knowing how to read their equations and find key points like the vertex can turn a scary-looking problem into a quick win.

Where this lesson fits

Nonlinear Functions

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→Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables →Equivalent Expressions →Linear Functions

A nonlinear function is any function whose graph is not a straight line. The most common one on the test is the quadratic function — a function with a squared term, like f(x) = x² + 3x - 4. Its graph is a U-shaped curve called a parabola.

There are three forms of a quadratic you should recognize:

Standard form: f(x) = ax² + bx + c. Here c is the y-intercept (where the curve crosses the y-axis), and the sign of a tells you which way the parabola opens. If a > 0, it opens up (a smile); if a < 0, it opens down (a frown).
Vertex form: f(x) = a(x - h)² + k. The point (h, k) is the vertex — the lowest point if the parabola opens up, or the highest point if it opens down. Watch the sign: (x - 3)² means h = 3, but (x + 3)² means h = -3.
Factored form: f(x) = a(x - r₁)(x - r₂). The values r₁ and r₂ are the x-intercepts (also called roots or zeros) — where the curve crosses the x-axis.

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The parabola f(x) = x² - 4x - 1, opening up with vertex at (2, -5).

The single most-tested skill is finding the vertex. Two reliable methods:

Formula: The x-coordinate of the vertex is x = -b / (2a). Plug that x back into the function to get the y-coordinate.
Completing the square to rewrite standard form into vertex form a(x - h)² + k.

The vertex x-coordinate also sits on the axis of symmetry — a vertical line that splits the parabola into mirror-image halves.

Besides quadratics, "nonlinear" includes exponential functions like f(x) = 2^x (which grow or shrink by a constant factor) and other curves. But on most questions, you'll be evaluating a function (plug in a number for x and simplify) or finding a key feature (vertex, intercept, or zero). Master those two moves and you've got most of the points.

When you evaluate, just substitute carefully and follow order of operations — square first, then multiply, then add.

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Quick check

Check your understanding with a question from this topic:

What is the vertex of the parabola f(x) = -x² + 2x + -5?

(-1, -4)

(1, 4)

(1, -4)

(2, -5)

Worked examples

Example 1

If f(x) = 3x² - 2x - 1, what is f(2)?

Example 2

What is the vertex of the parabola f(x) = x² - 4x - 1?

Example 3

The function f(x) = -x² + 2x - 5 is written in vertex form as f(x) = a(x - h)² + k. What is the value of k?

Common pitfalls

Forgetting the negative in -b/(2a)

The vertex formula has a minus sign. If b = -4, then -b = +4, not -4. Sign errors here flip your whole answer — write out -(-4) explicitly.

Misreading the sign of h in vertex form

In a(x - h)², the vertex x-coordinate is +h. So (x - 3)² gives h = 3, but (x + 3)² means (x - (-3))², so h = -3. The sign inside flips.

Wrong order of operations when evaluating

In 3x², you square x before multiplying by 3. So 3(2)² = 3(4) = 12, not (3·2)² = 36. Square first, then multiply.

Confusing zeros, y-intercept, and vertex

Zeros are where y = 0 (x-intercepts), the y-intercept is where x = 0, and the vertex is the turning point. Read what the question actually asks for before solving.

Key takeaways

A nonlinear function's graph isn't a straight line; the most common is the quadratic, whose graph is a parabola.
Vertex x-coordinate = -b/(2a); substitute it back into f(x) to get the y-coordinate.
Vertex form a(x-h)²+k reveals the vertex (h,k) directly — watch the sign of h.
If a > 0 the parabola opens up (minimum); if a < 0 it opens down (maximum).
To evaluate f at a number, substitute and follow order of operations: square first, then multiply, then add.

Watch & learn

Curated Khan Academy walkthroughs on Nonlinear Functions. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

Tracks your progress across lessons.

Try it yourself

5 practice questions on Nonlinear Functions, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Nonlinear Functions

You are here

→Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables →Equivalent Expressions →Linear Functions