Quadratics Reference

Quadratic Functions

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.

Solved Examples

Jump directly to a worked example by problem type.

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 $f(x)=x^2-4x+3$.

Solution Steps
  1. Use the axis formula with $a=1$ and $b=-4$.
  2. Substitute $x=2$ into the function.
  3. Evaluate the function at the axis value.
  4. State the vertex: $(2,-1)$.
  5. State the axis of symmetry: $x=2$.
$$h=-\frac{-4}{2(1)}=2$$ $$k=f(2)=(2)^2-4(2)+3$$ $$k=-1$$ $$\text{vertex: }(2,-1)$$ $$\text{axis: }x=2$$

Graph from Vertex Form

Describe $g(x)=-(x-2)^2+3$.

Solution Steps
  1. Read the vertex-form equation and its coefficient.
  2. State the vertex: $(2,3)$.
  3. Since the coefficient is negative, the parabola opens downward.
  4. Identify the reflection over the x-axis.
  5. State the maximum value: $3$.
$$g(x)=-(x-2)^2+3$$ $$\text{vertex: }(2,3)$$ $$\text{opens downward}$$ $$\text{reflection over the x-axis}$$ $$\text{maximum value: }3$$

Intercepts from Factored Form

Find the x-intercepts and axis of $y=(x-1)(x-3)$.

Solution Steps
  1. Set each factor equal to zero.
  2. State the x-intercepts: $(1,0)$ and $(3,0)$.
  3. Average the two x-intercepts to find the axis.
  4. State the axis of symmetry: $x=2$.
$$x=1,\quad x=3$$ $$\text{x-intercepts: }(1,0),\ (3,0)$$ $$x=\frac{1+3}{2}=2$$ $$\text{axis: }x=2$$