Quadratic Functions: Practice Problems and Methods

Completing the square: vertex and axis

Example: $y = 2x^2 - 8x + 5 = 2(x - 2)^2 - 3$, so the vertex is $(2,\ -3)$ and the axis is $x = 2$.

A translation by $p$ in the $x$-direction and $q$ in the $y$-direction moves the vertex by $(p,\ q)$. Reflecting in the $x$-axis replaces $y$ with $-y$; reflecting in the $y$-axis replaces $x$ with $-x$.

Maximum and minimum values

Over all real numbers, the vertex gives the minimum if $a > 0$ and the maximum if $a < 0$. On an interval, check whether the vertex is inside and compare the values at the endpoints.

To find a quadratic, write $y = a(x - p)^2 + q$ if the vertex is known, $y = a(x - \alpha)(x - \beta)$ if the $x$-intercepts are known, and $y = ax^2 + bx + c$ otherwise, then substitute the conditions.

Max and min with a parameter

The minimum of $f(x) = x^2 - 2ax + 3$ on $0 \leqq x \leqq 2$ depends on where the axis $x = a$ lies:

• $a < 0$ (axis left of the interval): minimum $f(0) = 3$
• $0 \leqq a \leqq 2$ (axis inside): minimum $f(a) = -a^2 + 3$
• $a > 2$ (axis right of the interval): minimum $f(2) = 7 - 4a$

For the maximum, compare the axis with the midpoint $x = 1$. When the interval itself moves, as in $a \leqq x \leqq a + 2$, reason the same way about the axis and the interval.