Skill: Interpret key features of a function from its graph

A graph shows how the output $y$ depends on the input $x$.

• The domain is the set of possible $x$-values.
• The range is the set of possible $y$-values.
• An intercept is where the graph crosses an axis.
• A zero is an $x$-value where $f(x)=0$, so the graph crosses or touches the $x$-axis.
• To describe increasing or decreasing, move from left to right on the graph.

Worked example

Suppose a graph is made of straight-line segments connecting these points:

$$(-4,2),\quad (-2,-2),\quad (0,0),\quad (2,4),\quad (4,1)$$

Assume all the endpoints are included.

1. Find the domain

Look at the leftmost and rightmost $x$-values.

• The graph begins at $x=-4$.
• The graph ends at $x=4$.

Therefore, the domain is

$$\boxed{[-4,4]}$$

2. Find the range

Look at the lowest and highest $y$-values.

• The lowest output is $y=-2$.
• The highest output is $y=4$.

Therefore, the range is

$$\boxed{[-2,4]}$$

3. Find the zeros and intercepts

The graph crosses the $x$-axis when $y=0$.

From the graph:

• It crosses at $(-3,0)$.
• It also passes through $(0,0)$.

So the zeros are

$$\boxed{x=-3\text{ and }x=0}$$

The $x$-intercepts are

$$\boxed{(-3,0)\text{ and }(0,0)}$$

The graph crosses the $y$-axis when $x=0$. Therefore, the $y$-intercept is

$$\boxed{(0,0)}$$

Remember that an intercept is written as a point, while a zero is written as an $x$-value.

4. Describe where the function increases and decreases

Read the graph from left to right.

• From $x=-4$ to $x=-2$, the graph goes downward, so the function is decreasing.
• From $x=-2$ to $x=2$, the graph goes upward, so the function is increasing.
• From $x=2$ to $x=4$, the graph goes downward again, so the function is decreasing.

Thus:

$$\boxed{\text{Decreasing on }[-4,-2]\text{ and }[2,4]}$$ $$\boxed{\text{Increasing on }[-2,2]}$$

There are no horizontal parts, so the function is not constant on any interval.

5. Find where the function is positive or negative

The function is positive where the graph is above the $x$-axis:

$$\boxed{[-4,-3)\cup(0,4]}$$

The function is negative where the graph is below the $x$-axis:

$$\boxed{(-3,0)}$$

The zeros $x=-3$ and $x=0$ are not included in the positive or negative intervals because $f(x)=0$ there.

6. Identify relative extrema

A relative minimum occurs where the graph changes from decreasing to increasing.

• At $(-2,-2)$, the graph changes from decreasing to increasing.
• Therefore, there is a relative minimum at

$$\boxed{(-2,-2)}$$

A relative maximum occurs where the graph changes from increasing to decreasing.

• At $(2,4)$, the graph changes from increasing to decreasing.
• Therefore, there is a relative maximum at

$$\boxed{(2,4)}$$

The coordinates are important: $-2$ is the input where the minimum occurs, and $-2$ is also the output in this example. In general, the input is the $x$-coordinate and the output is the $y$-coordinate.

7. Check for other features

This graph has:

• No breaks or holes, so it has no visible discontinuities.
• No clear symmetry.
• No end behavior to describe beyond $x=-4$ and $x=4$, because the graph is only shown on the domain $[-4,4]$.

When reading an actual graph, report values as estimates if the graph does not give exact coordinates. For example, you might write $f(1)\approx 2$ if the graph appears to have height about $2$ when $x=1$.