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Nonlinear Functions for the SAT: Complete Study Guide + Free Practice Problems
Everything the SAT tests about nonlinear functions in one place: quadratic functions and parabola features, exponential growth and decay, comparing growth rates, transformations, end behavior, and interpreting graphs in context — with step-by-step examples, worked problems, and 4 free practice quizzes.
Practice these quizzes
Free · No signupNonlinear Functions 1
Parabola features: vertex, axis of symmetry, intercepts.
Start quiz → Quiz 2Nonlinear Functions 2
Exponential growth and decay functions.
Start quiz → Quiz 3Nonlinear Functions 3
Comparing growth rates and transformations.
Start quiz → Quiz 4Nonlinear Functions 4
End behavior, domain/range, and graph interpretation.
Start quiz →What is a nonlinear function?
A nonlinear function is any function whose graph is not a straight line — the rate of change is not constant. On the SAT, this category is dominated by two families: quadratic functions (graphs shaped like a parabola) and exponential functions (graphs that grow or shrink by a constant multiplier).
| Function type | Form | Graph shape |
|---|---|---|
| Quadratic | f(x) = ax² + bx + c | Parabola (U-shaped curve) |
| Exponential | f(x) = a · bₓ | Curve that grows/shrinks faster and faster |
To confirm a table represents a nonlinear function, check that the differences between consecutive y-values are not constant. If those differences themselves grow at a constant rate, the table is quadratic; if they multiply by a constant factor, it's exponential.
Quadratic functions & parabola features
Every parabola has a few key features the SAT tests directly, and the form of the equation you're given often reveals one feature immediately without any extra algebra.
Vertex
The maximum or minimum point of the parabola. Easiest to read from vertex form: f(x) = a(x−h)²+k → vertex is (h, k).
Axis of symmetry
The vertical line x = h that splits the parabola into two mirror halves, passing through the vertex.
Direction of opening
If a > 0, the parabola opens upward (minimum). If a < 0, it opens downward (maximum).
x-intercepts (roots)
Easiest to read from factored form: f(x) = a(x−p)(x−q) → roots are p and q.
What is the vertex of f(x) = 2(x − 3)² + 5?
In vertex form, the h-value is subtracted inside the parentheses. If the equation shows (x + 3)², that means h = −3, not 3 — watch the sign carefully.
Exponential growth & decay
An exponential function changes by a constant percentage or factor for every unit increase in x, rather than a constant amount.
A population starts at 500 and increases by 8% each year. Write a function for population P after t years.
For decay, subtract the percentage from 100%. A quantity decreasing by 8% each year uses b = 1 − 0.08 = 0.92, not 0.08.
Comparing linear, quadratic & exponential growth
The SAT sometimes asks you to compare how different function types grow over the long run. The ranking is always the same once x gets large enough.
| Function type | Growth pattern |
|---|---|
| Linear | Adds the same amount every step |
| Quadratic | Grows faster than linear, but slower than exponential in the long run |
| Exponential | Eventually outgrows every polynomial function, no matter the coefficients |
Even an exponential function with a small growth factor (like 1.01ₓ) will eventually overtake any quadratic function, no matter how large its coefficients are — it just takes a larger x-value to see it happen.
Graph transformations
Adding or multiplying constants onto a parent function shifts, stretches, or reflects its graph in predictable ways.
| Change to f(x) | Effect on the graph |
|---|---|
| f(x) + k | Shifts up (k > 0) or down (k < 0) |
| f(x − h) | Shifts right (h > 0) or left (h < 0) |
| a · f(x), |a| > 1 | Stretches vertically (steeper/taller) |
| a · f(x), 0 < |a| < 1 | Compresses vertically (flatter/shorter) |
| −f(x) | Reflects over the x-axis |
Horizontal shifts move opposite to the sign inside the parentheses. f(x − 3) shifts the graph right by 3, not left, even though the sign is negative.
End behavior, domain & range
End behavior describes what happens to f(x) as x moves toward positive or negative infinity. Domain and range describe every possible input and output value.
What is the range of f(x) = −(x − 2)² + 9?
For any parabola, the domain is always all real numbers — only the range is restricted, and it's restricted by the vertex's y-value in the direction the parabola opens.
Interpreting graphs in context
The SAT often shows a nonlinear graph modeling a real situation — height of a projectile, value of an investment, spread of a population — and asks what a specific feature means.
- Vertex of a projectile graph: usually the maximum height and the time it occurs.
- y-intercept: the starting value at time zero — initial height, initial investment, initial population.
- x-intercepts of a height graph: the times when the object is at ground level (launch and landing).
- Where the exponential graph crosses a horizontal line: the time at which a quantity reaches a specific target value.
Test day strategy for nonlinear functions
| Question signal | Fastest approach |
|---|---|
| Vertex or max/min asked | Look for vertex form first; complete the square if given standard form |
| x-intercepts asked | Look for factored form first; factor standard form if needed |
| "Increases by x% each year/period" | Growth factor = 1 + rate; decay factor = 1 − rate |
| Comparing two function types long-term | Exponential eventually beats quadratic, which eventually beats linear |
| Graph shifted on the coordinate plane | Match the shift to the transformation rules — check signs carefully |
| Range of a parabola | Identify the vertex y-value and the direction the parabola opens |
Now put it to work
Four quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Nonlinear Functions 1
Parabola features: vertex, axis of symmetry, intercepts.
Start quiz → Quiz 2Nonlinear Functions 2
Exponential growth and decay functions.
Start quiz → Quiz 3Nonlinear Functions 3
Comparing growth rates and transformations.
Start quiz → Quiz 4Nonlinear Functions 4
End behavior, domain/range, and graph interpretation.
Start quiz →