Quadratics Reference
Quadratic Graphs and Parabolas
A quadratic function graphs as a parabola. The most useful form depends on the question: standard form shows the y-intercept, vertex form shows transformations, and factored form shows x-intercepts.
Fact Table
| Form | What It Shows | Use It For |
|---|---|---|
| $y=ax^2+bx+c$ | Standard form; y-intercept is $c$. | Discriminant, axis formula, quick y-intercept. |
| $y=a(x-h)^2+k$ | Vertex form; vertex is $(h,k)$. | Graph transformations, maximum or minimum. |
| $y=a(x-r_1)(x-r_2)$ | Factored form; x-intercepts are $r_1$ and $r_2$. | Zeros, intercept sketches, sign behavior. |
| $a>0$ | Opens upward. | The vertex is a minimum. |
| $a<0$ | Opens downward. | The vertex is a maximum. |
Content Formulas
Axis of Symmetry
$$x=-\frac{b}{2a}$$
Vertex from Standard Form
$$h=-\frac{b}{2a},\quad k=f(h)$$
Vertex Form
$$y=a(x-h)^2+k$$
Factored Form
$$y=a(x-r_1)(x-r_2)$$
For graphing, find the vertex first. Then use symmetry, intercepts, or one point on each side to place the parabola cleanly.
Classic Examples
Vertex from Standard Form
Find the vertex and axis of symmetry for $f(x)=2x^2-8x+3$.
Solution StrategyUse $x=-\frac b{2a}$, then substitute to find the y-coordinate.
$$h=-\frac{-8}{2(2)}=2$$
$$k=f(2)=2(2)^2-8(2)+3$$
$$k=-5$$
$$\text{vertex: }(2,-5)$$
$$\text{axis: }x=2$$
Graph from Vertex Form
Describe $g(x)=-3(x+1)^2+4$.
Solution StrategyRead the vertex and transformations directly from vertex form.
$$g(x)=-3(x-(-1))^2+4$$
$$\text{vertex: }(-1,4)$$
$$\text{opens downward}$$
$$\text{vertical stretch by }3$$
$$\text{maximum value: }4$$
Intercepts from Factored Form
Find the x-intercepts and axis of $y=-(x-2)(x+6)$.
Solution StrategyThe zeros are visible. The axis is halfway between them.
$$x=2,\quad x=-6$$
$$\text{x-intercepts: }(2,0),\ (-6,0)$$
$$x=\frac{2+(-6)}{2}=-2$$
$$\text{axis: }x=-2$$