Key Features of Functions

Every function graph can be described by a short list of features: where it crosses the axes, its highest or lowest points, where it rises or falls, and what it does at the far edges. These are the vocabulary of Algebra 2.

Reading these features lets you compare functions and sketch them quickly. This lesson names each one and shows how to pull it from a graph or a simple equation.

Intercepts and extrema

The $x$ -intercepts are where the graph meets the x-axis (y = 0); the $y$ -intercept is where it meets the y-axis (x = 0). A maximum is a peak, a minimum is a valley.

A downward parabola like $f(x) = -(x - 3)^2 + 7$ has a maximum at its vertex. In vertex form the vertex is $(3, 7)$ , so the maximum value is $7$ .

The downward parabola below shows the idea: it has a maximum at its vertex $(1, 4)$ , two $x$ -intercepts where it crosses the x-axis, and a $y$ -intercept where it crosses the y-axis.

Increasing, decreasing, and end behavior

A function increases where the graph rises left-to-right and decreases where it falls. These are stated as intervals of x.

End behavior describes what happens as x heads to $-\infty$ and $+\infty$ — does the graph rise, fall, or level off? It is the big-picture direction of the two tails.

Worked examples

Example 1: maximum from vertex form

Find the maximum value of $f(x) = -(x - 3)^2 + 7$ .

Negative leading coefficient opens down

\text{has a maximum}

Vertex form gives the vertex

(3, 7)

The maximum is the y-value

7

Answer: $7$

Example 2: intercepts

Find the $y$ -intercept of $f(x) = 2x - 6$ .

Set x = 0

f(0) = 2(0) - 6

Simplify

-6

Answer: $(0, -6)$

Example 3: where a function is increasing

On what interval is $f(x) = (x - 2)^2 - 5$ increasing?

Positive leading coefficient opens up

\text{has a minimum}

Vertex form gives the vertex

(2, -5)

The graph falls to the left of the vertex and rises to the right

x > 2

Answer: Increasing for $x > 2$

Try one yourself

Common questions

What is the difference between a maximum value and where it occurs?

The maximum value is the y-coordinate of the peak; where it occurs is the x-coordinate. For a vertex $(3, 7)$ , the maximum value is $7$ , occurring at $x = 3$ .

Can a function have more than one maximum?

Yes — higher-degree graphs can have several local peaks and valleys. A parabola has just one, but a quartic can have two valleys and a peak between them.

What does end behavior tell me?

It tells you the direction of the two tails as x gets very large or very small — useful for matching a graph to its equation's degree and leading sign.

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Key Features of Functions

Intercepts and extrema

Increasing, decreasing, and end behavior

Worked examples

Example 1: maximum from vertex form

Example 2: intercepts

Example 3: where a function is increasing

Try one yourself

Common questions

Keep learning

Domain & Range from Graphs and Equations

Symmetry: Even & Odd Functions

Sketching & Comparing Functions