Let’s start with defining a function. A function relates two variables (x, y), where y is dependent on x. For example, in the given function: y=3x, if x=2, then y=3(2) which is y=6. But, if x=4, then y=3(4) which is y=12. You see how multiplying 3 by different x-values gives us different y-values? This is what a function does, where x must be known to find y. So, how does this apply to graphing?
When graphing a function, the x-values are called input, and the y-values are called output. This is because the x-values Rae the values you’re inputting into the function which will output a y-coordinate. A function’s domain is the set of all x-values that satisfy the function, meaning they are the values that provide a y-coordinate. The range is the set of all y-values produced from the x-values, so the range is the set of outputs when x is a specific input. To graph a function, we graph points as an ordered pair. Remember, an ordered pair takes the form (x, y); it is the pair of an x-coordinate that relates a y-coordinate. So, we plot points on a coordinate plane. The idea is that x-coordinates, or input, are located first; these are the coordinates that shift laterally (left to right). Then, we locate the corresponding y-value, or input, for that x-value, and graph the point. So, we first travel along the x-axis, then rise or drop following the y-axis and plot the point.
So, let’s look at the function y=3x. We are given the domain: remember the domain is the set of all x-values. So, the domain values are the values we plug in for x. So, we will plug in -4 for x:
y=3(-4)
y=-12
Our point is (4, -12) because our input (x) is 4 and output (y) is -12. This means we travel a distance of 4 units along the x-axis, then down down units following the y-axis.
The process repeats for the remaining domain values:
y=3(-2)
y=-6
The point is (-2, -6), so we travel 2 units to the left (because the value is -2, not 2), then down 6 units.
y=3(0)
y=0
The point is (0, 0), so we travel no spaces left or right and no spaces up or down.
y=3(2)
y=6
The point is (2, 6), so we travel 2 units to the right (2 is a positive value) and 6 units up.
That’s how a function works! X-values are input into the equation and from there, you simplify and solve for y, the output. These points are graphed as a coordinate pair (x, y), so you travel lateral “x” units and vertically “y” units, then plot the point. Then, you connect the points with a line.