Quadratic Functions Model Curved Change

A quadratic function has the form $f(x)=ax^2+bx+c$ with $a\ne0$. Its rate of change is not constant: equal steps in $x$ produce linearly changing first differences and constant second differences.

Three forms, three views

Standard form $ax^2+bx+c$ exposes the vertical intercept $c$. Factored form $a(x-r_1)(x-r_2)$ exposes zeros. Vertex form $a(x-h)^2+k$ exposes the turning point $(h,k)$ and axis $x=h$.

Completing the square

The identity

$x^2+bx=\left(x+\frac b2\right)^2-\left(\frac b2\right)^2$

creates a perfect square without changing value. For a general quadratic, it leads to vertex coordinate

$h=-\frac{b}{2a}.$

If $a>0$, the vertex is a minimum; if $a<0$, it is a maximum.

Transformations

Relative to $y=x^2$, $a(x-h)^2+k$ shifts right by $h$, vertically by $k$, reflects across the horizontal axis when $a<0$, and changes vertical scale by $|a|$.

Check your understanding

Describe $f(x)=-2(x+1)^2+8$: vertex, axis, opening, and maximum value.

Show the reasoning

The vertex is $(-1,8)$, the axis is $x=-1$, it opens downward because $a=-2$, and its maximum value is $8$.