SAT Math 3: Quadratics and Nonlinear FunctionsSAT Math 3: Quadratics and Nonlinear Functions

Quadratics and Nonlinear Functions

Goal: Factor, solve, and interpret quadratic functions in standard, factored, and vertex forms.

THE SHORT VERSION

A quadratic function has a parabola for a graph. The SAT asks you to solve quadratics, find vertices and zeros, and use the discriminant to count solutions.

1 Three Forms of a Quadratic

Form Equation Easiest to read

Standard y = ax² + bx + c y-intercept c; vertex x = −b/(2a)

Factored y = a(x − r)(x − s) Zeros (x-intercepts) r and s

Vertex y = a(x − h)² + k Vertex (h, k); max or min value k

If a > 0 the parabola opens up (the vertex is a minimum). If a < 0 it opens down (a maximum).

The axis of symmetry is x = −b/(2a), halfway between the zeros.

Example: y = −2(x − 3)² + 8 has vertex (3, 8) and a maximum value of 8.

Roots, vertex, y-intercept and axis of symmetry on one graph.

2 Solving Quadratic Equations

Factoring: set to 0 first. x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or 3.

Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a).

Completing the square: rewrite as (x − h)² = k, then take square roots.

Useful patterns: a² − b² = (a − b)(a + b); a² + 2ab + b² = (a + b)².

Sum and product of roots: sum = −b/a, product = c/a.

3 The Discriminant (b² − 4ac)

Value Real solutions x-intercepts

Positive Two Graph crosses the x-axis twice

Zero One Graph touches the x-axis once (vertex on the axis)

Negative None (real) Graph never reaches the x-axis

Example: x² + 4x + k = 0 has exactly one solution when 16 − 4k = 0, so k = 4.

4 Function Notation and Transformations

f(3) means substitute 3 for x. If f(x) = x² − 2x, then f(3) = 9 − 6 = 3.

f(x) + k shifts up k; f(x − h) shifts right h; −f(x) reflects over the x-axis; a·f(x) stretches vertically. A zero of f is an x with f(x) = 0 (an x-intercept).

To find where two functions are equal, set them equal and solve (or find the intersection on a graph).

5 Check Yourself

Q: What are the solutions of x² − 9 = 0? A: x = 3 or −3.

Q: What is the minimum value of y = (x + 2)² − 5? A: −5, at x = −2.

Q: What is the vertex x-coordinate of y = x² − 8x + 3? A: −b/(2a) = 8/2 = 4.

6 Words to know

Zero / root

x-value where the function equals 0.

Vertex

Highest or lowest point of a parabola.

Discriminant

b² − 4ac; tells how many real solutions.

Axis of symmetry

Vertical line through the vertex.

Function notation

f(x) names the output for input x.

STRATEGY Use Desmos to find vertices and intercepts quickly, but know the algebra for questions with variables. TRAP Remember to set the equation equal to zero before factoring.

SAT Math Guide 3 · original study notes