Function Graphs and Families

Read graphs as input–output sets, identify domains, ranges, intercepts, and regional behavior, and recognize common function families. Predict how transformations shift, scale, and reflect a known graph.

A function graph places inputs on the horizontal axis and outputs on the vertical axis. Its shape can reveal where a function is positive or negative, increasing or decreasing, curved or flat, bounded or unbounded. These visual patterns complement the formula; they do not replace it.

A Graph Is a Set of Input–Output Points

For the function

f(x)=2x+1,

each allowed input $x$ produces one output $f(x)$ . The graph contains every point $(x,f(x))$ in the chosen domain. A few of those points are:

$x$	$f(x)$	Point on the graph
$-1$	$-1$	$(-1,-1)$
$0$	$1$	$(0,1)$
$1$	$3$	$(1,3)$
$2$	$5$	$(2,5)$

The domain is the set of allowed inputs, and the range is the set of outputs the function actually produces. A drawn graph shows only a viewing window, so the visible part may not show the function's complete domain or range.

The graph of $f(x)=2x+1$ crosses the vertical axis at $(0,1)$ . This is its vertical intercept, or $y$ -intercept. Its zero, also called an $x$ -intercept or root, is the input for which $f(x)=0$ :

2x+1=0 \quad\Longrightarrow\quad x=-\frac{1}{2}.

Thus the graph crosses the horizontal axis at $\left(-\frac{1}{2},0\right)$ . Each increase of $1$ in $x$ increases the output by $2$ , so the line has slope $2$ .

Exercise: Find a zero from a formula

At which input does $f(x)=3x-6$ cross the horizontal axis?

Your answer

Compute it first, then check your number.

HintSet the output to zero

Solve $3x-6=0$ .

SolutionSolve for the horizontal intercept

A zero is an input whose output is $0$ :

3x-6=0 \quad\Longrightarrow\quad 3x=6 \quad\Longrightarrow\quad x=2.

The corresponding point on the graph is $(2,0)$ .

Common Families Have Recognizable Structure

A function family is a collection of functions with a shared algebraic form. Parameters select one member of the family. Recognizing a family gives useful expectations, but its parameter values and domain still determine the particular graph.

Family	Typical form	Useful first observations
constant	$f(x)=c$	horizontal line; range contains one value
linear	$f(x)=mx$	passes through the origin; slope is $m$
affine	$f(x)=mx+b$	straight line; vertical intercept is $b$
quadratic	$f(x)=ax^2+bx+c$	parabola; opens upward or downward according to the sign of $a$
power	$f(x)=x^p$	symmetry and end behavior depend on the exponent $p$
exponential	$f(x)=b^x$	positive output; growth for $b>1$ and decay for $0<b<1$
logarithmic	$f(x)=\log_b x$	positive-input domain; inverse shape of $b^x$
piecewise	different rules on different regions	corners, jumps, or changes of behavior may occur at boundaries

The distinction between linear and affine is worth preserving. In formal linear algebra, $f(x)=mx$ is linear because it maps $0$ to $0$ . If $b\ne0$ , the rule $f(x)=mx+b$ is affine. Terms such as “linear regression” and “linear layer” often include an intercept or bias, so context matters.

A piecewise function states both its formulas and the regions where they apply. For example:

\operatorname{ReLU}(x)= \begin{cases} 0, & x<0,\\ x, & x\ge0. \end{cases}

Its graph is horizontal for negative inputs and follows the line $y=x$ for nonnegative inputs. The two pieces meet at the boundary $x=0$ .

Exercise: Recognize a quadratic rule

Which function is quadratic?

Choose one function

f(x) = 3x - 1

f(x) = x² + 2x

f(x) = 2ˣ

Select one choice, then check.

HintInspect the powers of the input

In a quadratic polynomial, the largest exponent of $x$ is $2$ .

SolutionIdentify the squared term

The rule $f(x)=x^2+2x$ is quadratic because its highest power of $x$ is $2$ . The rule $3x-1$ is affine, while $2^x$ is exponential because the variable appears in the exponent.

Read Behavior over Regions

A function is increasing on a region if larger inputs there produce larger outputs. It is decreasing if larger inputs produce smaller outputs. A function may do both on different regions. For example, $f(x)=x^2$ decreases as $x$ moves from negative values toward $0$ and increases for positive $x$ .

A graph can also show symmetry. The graph of $x^2$ is symmetric across the vertical axis because $(-x)^2=x^2$ . The graph of $x^3$ has rotational symmetry about the origin because $(-x)^3=-x^3$ .

Some functions are bounded above or below; their outputs cannot pass a fixed boundary. Others approach a line without reaching it. Such a line is an asymptote. For instance, $e^x$ approaches the horizontal axis as $x\to-\infty$ but remains positive, while $\ln x$ decreases without bound as $x$ approaches $0$ from the right.

Words such as curvature, saturation, maximum, and minimum describe more specific behavior. Later chapters will define these ideas more precisely. At this stage, the important habit is to name the region and the visible evidence rather than describing an entire function from one small part of its graph.

Transformations Move and Scale a Known Graph

Starting from $y=f(x)$ , changes outside the function alter outputs, while changes inside the function alter inputs:

New function	Effect on the graph of $f$
$f(x)+c$	shift vertically by $c$ ; upward when $c>0$
$f(x-c)$	shift horizontally to the right by $c$ when $c>0$
$a f(x)$	scale vertically by $\lvert a\rvert$ and reflect across the horizontal axis when $a<0$
$f(ax)$	compress horizontally when $\lvert a\rvert>1$ , stretch when $0<\lvert a\rvert<1$ , and reflect across the vertical axis when $a<0$
$-f(x)$	reflect across the horizontal axis
$f(-x)$	reflect across the vertical axis

For the horizontal scaling rule $f(ax)$ , assume $a\ne0$ . If $a=0$ , every input is sent to $f(0)$ , so the result is a constant function rather than a horizontally scaled copy of the original graph.

The horizontal shift can feel reversed. To find where $f(x-c)$ reproduces the old output $f(0)$ , solve $x-c=0$ , which gives $x=c$ . The feature formerly at $x=0$ therefore appears at $x=c$ , to the right when $c>0$ .

Exercise: Read a vertical shift

If $g(x)=f(x)+3$ , how does the graph of $g$ compare with the graph of $f$ ?

Choose the transformation

Every output is 3 larger, so the graph moves upward.

Every input is 3 larger, so the graph moves right.

Every output is multiplied by 3.

Select one choice, then check.

HintHold the input fixed

At any chosen $x$ , compare $g(x)$ directly with $f(x)$ .

SolutionCompare outputs at the same input

Since

g(x)=f(x)+3,

every output is exactly $3$ larger. Each point therefore moves upward by three units without changing its horizontal coordinate.

Exercise: Read a horizontal shift

If $h(x)=f(x-4)$ , where does a feature originally at input $x=0$ appear on the graph of $h$ ?

Your answer

Compute it first, then check your number.

HintMatch the old input

Solve $x-4=0$ to find where $h$ uses the old input $0$ .

SolutionSolve for the shifted input

The old feature occurred when the input to $f$ was $0$ . For $h$ that happens when:

x-4=0,

so $x=4$ . The graph shifts four units to the right.

Read the Axes before Reading the Shape

A graph can suggest zeros, boundaries, extrema, or long-term behavior, but its window and scale may hide important information. A reliable reading begins by naming the input and output, checking their units and displayed ranges, and identifying the function's domain. Then locate intercepts and boundaries, describe increasing or decreasing behavior on named regions, and verify important claims with the formula or numerical values.

This method applies to ordinary curves as well as later loss graphs, activation functions, probability densities, and optimization surfaces.