Connecting Representations of Functions

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Student Summary

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+2y = 3x +2, where yy is the dependent variable and xx is the independent variable. If we wanted to find the output that goes with 2, we could input 2 into the equation for xx and find the corresponding value of yy. In this case, when xx is 2, yy is 8 since 3⋅2+2=83\boldcdot 2 + 2=8.

If we had a graph of this function instead, then the coordinates of points on the graph would be the input-output pairs.

So we would read the yy-coordinate of the point on the graph that corresponds to a value of 2 for xx. Looking at the following graph of a function, we can see the point (2,8)(2,8) on it, so the output is 8 when the input is 2.

Coordinate plane, x, negative 1 to 2 by ones, y negative 2 to 8 by twos. Graph on a straight line through (0 comma 2), and (2 comma 8).
The graph of a line in the coordinate plane with the origin labeled “O”. The horizontal axis has the numbers negative 1 through 2 indicated and there are vertical gridlines between each integer. The vertical axis has the numbers negative 2 through 8, in increments of 2, indicated, and there are horizontal grid lines in between each integer. The line begins to the right of the y axis and below the x axis. It slants upward and to the right passing through the point with coordinates negative 1 comma negative 1, crosses the y axis at 2, and passes through the indicated point labeled 2 comma 8.

A table representing this function shows the input-output pairs directly (although only for select inputs).

Again, the table shows that if the input is 2, the output is 8.

xx -1 0 1 2 3
yy -1 2 5 8 11

Visual / Anchor Chart

Standards

Addressing
8.F.2

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

8.F.3

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. Recognize examples of functions that are linear and non-linear.