7.1: Which are the Same? Which are Different?
Here are three different ways of representing functions. How are they alike? How are they different?
 $y = 2x$

$p$ 2 1 0 1 2 3 $q$ 4 2 0 2 4 6
Let’s connect tables, equations, graphs, and stories of functions.
Here are three different ways of representing functions. How are they alike? How are they different?
$p$  2  1  0  1  2  3 

$q$  4  2  0  2  4  6 
The graph shows the temperature between noon and midnight in City A on a certain day.
The table shows the temperature, $T$, in degrees Fahrenheit, for $h$ hours after noon, in City B.
$h$  1  2  3  4  5  6 

$T$  82  78  75  62  58  59 
The volume, $V$, of a cube with side length $s$ is given by the equation $V = s^3$. The graph of the volume of a sphere as a function of its radius is shown.
Is the volume of a cube with side length $s=3$ greater or less than a sphere with radius 3?
Estimate the radius of a sphere that has the same volume as a cube with side length 5.
Compare the outputs of the two volume functions when the inputs are 2.
Here is an applet to use if you choose. Note: If you want to graph an equation with this applet, it expects you to enter $y$ as a function of $x$, so you need to use $y$ instead of $V$ and $x$ instead of $s$.
Elena’s family is driving on the freeway at 55 miles per hour.
Andre’s family is driving on the same freeway, but not at a constant speed. The table shows how far Andre's family has traveled, $d$, in miles, every minute for 10 minutes.
$t$  1  2  3  4  5  6  7  8  9  10 

$d$  0.9  1.9  3.0  4.1  5.1  6.2  6.8  7.4  8  9.1 
Functions are all about getting outputs from inputs. For each way of representing a function—equation, graph, table, or verbal description—we can determine the output for a given input.
Let's say we have a function represented by the equation $y = 3x +2$ where $y$ is the dependent variable and $x$ is the independent variable. If we wanted to find the output that goes with 2, we can input 2 into the equation for $x$ and finding the corresponding value of $y$. In this case, when $x$ is 2, $y$ is 8 since $3\boldcdot 2 + 2=8$.
If we had a graph of this function instead, then the coordinates of points on the graph are the inputoutput pairs. So we would read the $y$coordinate of the point on the graph that corresponds to a value of 2 for $x$. Looking at the graph of this function here, we can see the point $(2,8)$ on it, so the output is 8 when the input is 2.
A table representing this function shows the inputoutput pairs directly (although only for select inputs).
$x$  1  0  1  2  3 

$y$  1  2  5  8  11 
Again, the table shows that if the input is 2, the output is 8.