SAT Topic 2

Quadratic graphs

Vertex, intercepts, axis of symmetry, and which way the parabola opens.

Concept

The graph of y = ax² + bx + c is a parabola. Key features:

Direction — opens up if a > 0 (has a minimum), down if a < 0 (has a maximum).
Axis of symmetry — the vertical line x = −b / (2a). The parabola is mirror-symmetric across it.
Vertex — sits on the axis of symmetry. Its y is the function's max or min.
y-intercept — the constant term c (set x = 0).
x-intercepts — solve ax² + bx + c = 0.

Two useful equivalent forms: vertex form y = a(x − h)² + k with vertex (h, k), and intercept form y = a(x − p)(x − q) with x-intercepts at p and q.

Worked example 1

Find the vertex of `y = x² − 6x + 5`.

Solution

Axis. x = −b / (2a) = 6 / 2 = 3

y at vertex. y(3) = 9 − 18 + 5 = −4

Vertex: (3, −4). Since a > 0, this is a minimum point.

Worked example 2

Convert `y = (x − 1)(x + 5)` to vertex form.

Solution

Expand. y = x² + 4x − 5

Axis. x = −4 / 2 = −2

y at vertex. y(−2) = 4 − 8 − 5 = −9

Vertex form: y = (x + 2)² − 9. Vertex at (−2, −9).

Practice test

8 questions on vertex, axis of symmetry, intercepts, vertex form, and direction of opening.

Practice test · 8 questions Question 1 of 8 · Score 0