Reading Graphs and Plotting Points, foundations for calculus lesson

A weather app shows a line of temperatures across the day, and a fitness tracker shows distance against time. Both pictures are graphs, and each point on them is a pair of numbers with a meaning. Calculus studies the slopes of such pictures and the areas under them, so it pays to know exactly what a point says, what the scale on each axis is, and what a picture made from a few measurements does not tell you.

Key terms

Coordinate axes

Definition The coordinate plane is built from two perpendicular real number lines, the coordinate axes, that cross at their zero points. The crossing point is the origin. The horizontal axis increases to the right and the vertical axis increases upward. Each axis has its own unit length and is labeled with the quantity it measures and, in an applied problem, its unit.

In plain words Two number lines, one across and one up, meeting at zero. Their labels tell you what each direction measures.

Example To graph a cart’s position over time, label the horizontal axis “time $t$ in seconds” and the vertical axis “position $s$ in meters”. The origin then stands for time $0$ and position $0$ .

Ordered pair

Definition An ordered pair $(a, b)$ is a pair of real numbers in which order matters, so $(a, b) = (c, d)$ exactly when $a = c$ and $b = d$ . In the coordinate plane each point corresponds to exactly one ordered pair and each ordered pair to exactly one point. The point $(a, b)$ is found by moving $\lvert a \rvert$ units from the origin horizontally, to the right when $a > 0$ and to the left when $a < 0$ , and then $\lvert b \rvert$ units vertically, up when $b > 0$ and down when $b < 0$ .

In plain words A point is an address with two parts, written in a fixed order. Swapping the parts gives a different address.

Example The points $(2, 5)$ and $(5, 2)$ are different. The first is $2$ units right and $5$ up, the second is $5$ right and $2$ up. The point $(-3, 0)$ lies on the horizontal axis, three units left of the origin.

Horizontal and vertical coordinates

Definition In the ordered pair $(a, b)$ the first number $a$ is the horizontal coordinate, also called the $x$ -coordinate, and the second number $b$ is the vertical coordinate, also called the $y$ -coordinate. The horizontal coordinate is always listed first. Points on the vertical axis have horizontal coordinate $0$ , and points on the horizontal axis have vertical coordinate $0$ .

In plain words Across first, then up or down.

Example For the point $(-4, 3)$ the horizontal coordinate is $-4$ , so it lies left of the vertical axis, and the vertical coordinate is $3$ , so it lies above the horizontal axis. To read a point off a graph, drop a vertical line to the horizontal axis for the first coordinate and run a horizontal line to the vertical axis for the second.

Input and output on a graph

Definition When a function $f$ is graphed, its input is measured along the horizontal axis and its output along the vertical axis. A point $(a, b)$ lies on the graph exactly when $a$ is in the domain of $f$ and $b = f(a)$ . So the point of the graph at the input $a$ has vertical coordinate $f(a)$ , its height above the horizontal axis, or its depth below it when $f(a) < 0$ .

In plain words Go along the horizontal axis to your input, then straight up or down to the graph. How high you are is the output.

Example Let $s(t) = 2t^2$ be a cart’s position in meters $t$ seconds after it starts, used only for $t \ge 0$ . At the input $t = 3$ the output is $s(3) = 18$ , so the point $(3, 18)$ is on the graph. It says that at $3$ seconds the cart is $18$ meters from its start. The point $(18, 3)$ would wrongly claim a position of $3$ meters at $18$ seconds.

Graph of a function

Definition The graph of a function $f$ with domain $D$ is the set of points

\{(x, f(x)) \mid x \in D\}.

Because each input has exactly one output, every vertical line $x = a$ meets the graph of a function at most once, and it meets it exactly once when $a$ is in $D$ .

In plain words The graph is the picture of every input paired with its output. Above each input there is only one point.

Example For the cart, $s(t) = 2t^2$ with $t \ge 0$ . A table of values gives points to plot.

$t$ in seconds	$0$	$1$	$2$	$3$	$4$
$s(t)$ in meters	$0$	$2$	$8$	$18$	$32$

The points $(0, 0)$ , $(1, 2)$ , $(2, 8)$ , $(3, 18)$ and $(4, 32)$ lie on the graph. Here the formula is known, so the curve between them is known too, and $s(1.5) = 4.5$ puts $(1.5, 4.5)$ on it. The graph starts at $t = 0$ and has no points to the left, since negative times are not part of this model.

Scale

Definition The scale of an axis is the amount of the measured quantity represented by the distance between neighboring tick marks, or by one unit of length on the page. The two axes may have different scales. An ordinary axis uses the same scale along its whole length, so equal distances on it mean equal changes in the quantity. Because of this, how steep a graph looks depends on the scales chosen. The slope between two points is computed from their coordinates, as change in output divided by change in input, with units, never read from apparent steepness alone.

In plain words Before reading a graph, read the axes. A line can look steep or flat just from how the axes were stretched.

Example For the cart, a horizontal scale of $1$ second per tick and a vertical scale of $5$ meters per tick fit the table on a small grid. From $t = 1$ to $t = 2$ the position rises by $8 - 2 = 6$ meters, so the average rate is

\frac{8 - 2}{2 - 1} = 6 \text{ meters per second.}

Now keep the tick marks the same distance apart on the page but let each vertical tick stand for $1$ meter. The rise of $6$ meters then covers $6$ ticks instead of $1.2$ , five times as tall on the page over the same horizontal run, so the curve looks much steeper. The rate is still $6$ meters per second. If the ticks were also spaced differently, the picture would change by yet another factor, which is why the slope comes from the coordinates and not from the drawing.

Data points and the gaps between them

Definition A graph made from a finite table of measurements consists only of the listed points. The table does not determine any value at an unlisted input. Joining the points with segments or a curve, or estimating a value between them, adds an assumption, such as a formula or steady change, and the result is an estimate rather than data.

In plain words Measurements tell you what happened at the times you measured. Between them you are guessing, and you should say so.

Example Water temperature in a pot is recorded every two minutes.

Time in minutes	$0$	$2$	$4$	$6$
Temperature in $^\circ\text{C}$	$20$	$34$	$47$	$58$

The table gives the points $(0, 20)$ , $(2, 34)$ , $(4, 47)$ and $(6, 58)$ . It does not say the temperature at $3$ minutes. If you assume the temperature changed at a steady rate between $2$ and $4$ minutes, the midpoint gives an estimate of $40.5\,^\circ\text{C}$ , but that number comes from the assumption, not from the measurements.

Check yourself

Plot the formula $y = 3 - x$ at the inputs $-1$ , $0$ , $2$ and $4$ . List the points.

Show the answer

The outputs are $4$ , $3$ , $1$ and $-1$ , so the points are $(-1, 4)$ , $(0, 3)$ , $(2, 1)$ and $(4, -1)$ . Each input is listed first.

A graph shows a runner’s distance $d$ in meters against time $t$ in seconds, and it passes through $(10, 45)$ and $(20, 95)$ . What do these points mean, and what is the average speed between them?

Show the answer

At $10$ seconds the runner has covered $45$ meters, and at $20$ seconds $95$ meters. The average speed is

\frac{95 - 45}{20 - 10} = 5 \text{ meters per second.}

Using the temperature table above, can you say for certain whether the water was ever hotter than $58\,^\circ\text{C}$ between $0$ and $6$ minutes?

Show the answer

No. The table only gives the values at $0$ , $2$ , $4$ and $6$ minutes. Without a rule or an assumption about the heating, nothing is known about the temperature between those times.

Reading Graphs and Plotting Points

Key terms

Coordinate axes

Ordered pair

Horizontal and vertical coordinates

Input and output on a graph

Graph of a function

Scale

Data points and the gaps between them

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus