Skill: Determine intervals where a function is increasing or decreasing

Start by looking for where the function changes direction. A function is:

• increasing when larger $x$-values give larger $y$-values;
• decreasing when larger $x$-values give smaller $y$-values;
• constant when the $y$-value stays the same.

The points where the function changes direction are called turning points. These points separate the intervals.

Example

Determine the intervals where

$$f(x)=(x-2)^2-3,\qquad -1\le x\le 5$$

is increasing or decreasing.

Step 1: Identify the shape and turning point

The equation is in vertex form:

$$f(x)=(x-h)^2+k$$

Here, $h=2$ and $k=-3$, so the vertex is

$$(2,-3).$$

Because the squared term is positive, the parabola opens upward. Therefore, the vertex is a minimum point:

• as $x$ moves toward $2$ from the left, the function goes down;
• as $x$ moves right from $2$, the function goes up.

Step 2: State the intervals

The domain is $[-1,5]$, so the function is:

• decreasing on $[-1,2]$;
• increasing on $[2,5]$.

The endpoints are included because the given domain includes $-1$ and $5$.

Step 3: Check with a few values

$$\begin{array}{c|ccccc} x & -1 & 0 & 1 & 2 & 3 & 4 & 5\\ \hline f(x) & 6 & 1 & -2 & -3 & -2 & 1 & 6 \end{array}$$

From $x=-1$ to $x=2$, the output values go down:

$$6,\ 1,\ -2,\ -3$$

From $x=2$ to $x=5$, the output values go up:

$$-3,\ -2,\ 1,\ 6$$

Therefore,

$$\boxed{\text{Decreasing on }[-1,2]}$$

and

$$\boxed{\text{Increasing on }[2,5]}.$$

The point $x=2$ is the turning point, not a separate interval.