AP Precalculus: Quadratic Functions Formulas & Concepts

Quadratic Function Standard Form

General form: \( f(x) = ax^2 + bx + c \)

\( a \neq 0 \)
Opens up if \( a > 0 \), down if \( a < 0 \)
Parabola’s axis of symmetry: \( x = -\frac{b}{2a} \)

Maximum / Minimum Value

Vertex formula:
\( x_\text{vertex} = -\frac{b}{2a} \)
\( y_\text{vertex} = f(-\frac{b}{2a}) \)
If \( a > 0 \): minimum point; If \( a < 0 \): maximum point

Graphing a Quadratic

Shape: parabola
Vertex: \( \left(-\frac{b}{2a}, f(-\frac{b}{2a}) \right) \)
Axis of symmetry: \( x = -\frac{b}{2a} \)
Y-intercept: \( c \)
X-intercepts: Solve \( ax^2 + bx + c = 0 \) (roots)

Matching Quadratic Functions & Graphs

If \( a > 0 \), opens up; if \( a < 0 \), opens down
Width determined by \( |a| \): larger \( |a| \) = narrower parabola
\( b \) shifts vertex side-to-side; \( c \) is the y-intercept
Identify with vertex, axis, intercepts

Solving Quadratics by Square Roots

If \( ax^2 = k \):
\( x = \pm\sqrt{\frac{k}{a}} \)

Solving by Factoring

\( ax^2 + bx + c = 0 \Rightarrow (dx+p)(ex+q)=0 \)
Set each factor to zero: \( dx + p = 0 \), \( ex + q = 0 \); solve for \( x \)

Completing the Square

Rearrange: \( x^2 + bx = -c \)
Add \( \left(\frac{b}{2}\right)^2 \) to both sides:
\( x^2 + bx + \left(\frac{b}{2}\right)^2 = -c + \left(\frac{b}{2}\right)^2 \)
Factor left: \( (x+\frac{b}{2})^2 = \text{right side} \)

Quadratic Formula

\( x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a} \)

Always works to solve \( ax^2 + bx + c = 0 \)
\( \pm \): get two solutions unless the radical is zero

Using the Discriminant

\( D = b^2 - 4ac \)

If \( D > 0 \): two real solutions
If \( D = 0 \): one real solution
If \( D < 0 \): two complex solutions

Quadratic Word Problems

- Identify quantities that fit \( ax^2 + bx + c \)
- Solutions may represent maxima/minima, intercepts, or times.