Graphs and Special Functions: Quick Revision

A brief note on Graphs and Special Functions. Read it before a test. Try each question at the end before you open its answer.

Parabolas

$y = ax^2 + bx + c$ has its vertex at $x = -\dfrac{b}{2a}$, and opens upward when $a > 0$. Completing the square, $a(x - h)^2 + k$, shows the vertex $(h, k)$ and the range.

Special functions

Moving a graph

$f(x - h) + k$ moves it $h$ right and $k$ up. $a\,f(x)$ stretches the heights. $f(-x)$ reflects it in the y-axis and $-f(x)$ in the x-axis.

Symmetry and direction

Even: $f(-x) = f(x)$, symmetric in the y-axis. Odd: $f(-x) = -f(x)$, symmetric about the origin. A parabola falls on one side of its vertex and rises on the other.

A worked example

Describe $y = 2|x - 1| - 4$.

1. The corner. $|x|$ moved $1$ right and $4$ down: $(1, -4)$.
2. The range. $[-4, \infty)$.
3. The zeros. $2|x - 1| = 4$, so $x - 1 = \pm 2$: $x = -1$ or $x = 3$.

Practice with new numbers

A program to try

The program checks the worked example.

Mistakes to avoid

• Moving the wrong way. $f(x + 2)$ moves the graph $2$ to the left.
• $[x]$ for negative numbers. $[-2.3] = -3$, not $-2$.
• A fractional part below zero. $\{x\}$ is never negative.
• Calling every function even or odd. Most are neither.

Check yourself

1. Is $x^6 - x^2$ even, odd or neither?

2. What is $[-7.2]$?

3. What is the vertex of $y = x^2 + 2x - 8$?

4. Solve $|x + 3| > 1$.