Mathfolis

Nonlinear Functions

Unit 2 · Advanced Math

What the SAT asks here

Nonlinear functions on the SAT mean quadratics and exponentials. You'll switch between standard, vertex, and factored forms of a parabola, build exponential growth-and-decay models from context, and identify how transformations move the parent graph. Expect 3–4 problems per test.

Three forms of a parabola

Standard — reveals y-intercept (0, c)
f(x)=ax2+bx+cf(x) = ax^2 + bx + c
Vertex — reveals vertex (h, k)
f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k
Factored — reveals x-intercepts r1, r2
f(x)=a(x−r1)(x−r2)f(x) = a(x - r_1)(x - r_2)

Key utilities

Vertex x-coordinate (from standard form)
x=−b2ax = -\frac{b}{2a}
Exponential model
f(x)=a⋅bxf(x) = a \cdot b^x

Transformations of y=f(x)y = f(x)

f(x) + k — shift up by k
f(x − h) — shift right by h
−f(x) — reflect over the x-axis
a · f(x) — vertical stretch by a
SAT Tip: If you need the vertex, complete the square or use x = −b/(2a). If you need the x-intercepts, factor or use the quadratic formula. Match the form to the question.
Caution: Growth rate vs. growth factor: 5% growth means factor 1.05, NOT 0.05. A common SAT distractor uses the wrong number for the base.
Type 1

Find vertex or intercepts

Convert into the form that reveals what the question asks for: vertex form for (h, k), factored form for x-intercepts, standard form for the y-intercept.

Example 1
Find the vertex of f(x) = x² − 6x + 11.

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Type 2

Exponential growth and decay

Build a · bxb^x from context. The base b > 1 for growth, 0 < b < 1 for decay. If something doubles every k units, the factor is 2 per k units, i.e. 2(t/k)2^{(t/k)}.

Example 2
A population of 800 bacteria doubles every 3 hours. Write an expression for the population P(t) after t hours.

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Type 3

Function transformations

Read shifts, reflections, and stretches off the equation g(x) = a · f(x − h) + k.

Example 3
The graph of g(x) = f(x − 4) + 2 is the graph of f shifted how?

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