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SAT Math Unit 4: Nonlinear functions & equations

Advanced Math covers quadratics, exponentials and other nonlinear models. Know the forms and what each part means.

Covers: Quadratics and vertex form, exponential growth and decay, radical equations, nonlinear systems, function notation

Study guide

Nonlinear functions don't have a constant rate of change. Quadratics are the most common. Standard form y = ax² + bx + c shows the y-intercept c; vertex form y = a(x − h)² + k shows the vertex (h, k); factored form y = a(x − r)(x − s) shows the x-intercepts. The parabola opens up if a > 0 and down if a < 0, and its axis of symmetry is x = −b/(2a). Solve quadratics by factoring, by square roots, or with the quadratic formula. The discriminant b² − 4ac tells how many real solutions there are.

Exponential functions multiply by a constant factor each period. Growth is y = a(1 + r)ᵗ and decay is y = a(1 − r)ᵗ, where a is the starting value and r is the percent rate as a decimal. A table where each output is the previous one times the same number is exponential.

Radical equations are solved by isolating the radical and squaring both sides, which can create extraneous solutions, so always check answers in the original. Nonlinear systems, like a line and a parabola, are solved by substitution, and the number of intersection points is the number of solutions. Function notation f(x) means "the output when the input is x"; to evaluate, substitute, and to compose, work from the inside out.

If b² − 4ac > 0 there are two real solutions, if it equals 0 there is exactly one, and if it is negative there are no real solutions. For a line and a parabola, substitute to get one quadratic equation; its discriminant tells you whether they intersect twice, touch once or never meet. Because a parabola is symmetric, the vertex lies halfway between the x-intercepts: for y = a(x − r)(x − s), the axis of symmetry is x = (r + s)/2. Substitute that x-value to get the minimum or maximum value.

If a quantity doubles every d time units, it can be modeled as y = a·2^(t/d); if it halves every d units, use y = a·(1/2)^(t/d). The exponent t/d counts how many doubling or halving periods have passed. For ax² + bx + c = 0, the two solutions add to −b/a and multiply to c/a. This lets you find the sum of solutions without solving the equation.

Key ideas

• Vertex form y = a(x − h)² + k shows the vertex (h, k).
• Factored form shows x-intercepts; standard form shows the y-intercept c.
• The discriminant tells you how many times a parabola meets the x-axis (or a line).
• y = a(1 + r)ᵗ grows by r per period; y = a(1 − r)ᵗ decays.
• Square both sides to solve radical equations, then check for extraneous answers.

Formulas and rules

Common mistakes

• Keeping extraneous solutions after squaring.
• Reading 0.95 as 95% decay instead of 5% decay.

Words to know

Vertex of y = a(x − h)² + k

(h, k)

x-coordinate of the vertex of y = ax² + bx + c

−b/(2a)

Exponential decay model

y = a(1 − r)ᵗ: starts at a and shrinks by rate r each period.

Discriminant

b² − 4ac: positive → 2 real roots, zero → 1, negative → none real.

Extraneous solution (radicals)

A result from squaring that doesn't satisfy the original equation.

Exponential growth model

y = a(1 + r)ᵗ: starts at a and grows by rate r each period.

Vertex form

y = a(x − h)² + k, with vertex (h, k).

Factored form (quadratic)

y = a(x − r)(x − s); r and s are the x-intercepts.

Axis of symmetry

The vertical line x = −b/(2a) through a parabola's vertex.

Quadratic formula

x = [−b ± √(b² − 4ac)]/(2a)

Nonlinear system

A system with at least one nonlinear equation; solutions are intersection points.

Function notation

f(x) is the output for input x; f(3) means substitute 3.

Sample practice questions

469 more questions like these, from level 1 to 5, are in the practice set.