3.9 Function concepts -- graphing

Graphing, like algebraic generalizations, is a difficult topic because many students know how to do it but are not sure what it means .

For instance, consider the following graph:

If I asked you “Draw the graph of $y=x^2$ ” you would probably remember how to plot points and draw the shape.

But suppose I asked you this instead: “Here’s a function, $y=x^2$ . And here’s a shape, that sort of looks like a U. What do they actually have to do with each other?” This is a harder question! What does it mean to graph a function?

The answer is simple, but it has important implications for a proper understanding of functions. Recall that every point on the plane is designated by a unique $(x,y)$ pair of coordinates: for instance, one point is $(\mathrm{5,3})$ . We say that its $x$ -value is 5 and its $y$ -value is 3.

A few of these points have the particular property that their $y$ -values are the square of their $x$ -values. For instance, the points $(\mathrm{0,0})$ , $(\mathrm{3,9})$ , and $(-\mathrm{5,}25)$ all have that property. $(\mathrm{5,3})$ and $(-\mathrm{2,}-4)$ do not.

The graph shown—the pseudo-U shape—is all the points in the plane that have this property . Any point whose $y$ -value is the square of its $x$ -value is on this shape; any point whose $y$ -value is not the square of its $x$ -value is not on this shape. Hence, glancing at this shape gives us a complete visual picture of the function $y=x^2$ if we know how to interpret it correctly .

Graphing functions

Remember that every function specifies a relationship between two variables. When we graph a function, we put the independent variable on the $x$ -axis, and the dependent variable on the $y$ -axis.

For instance, recall the function that describes Alice’s money as a function of her hours worked. Since Alice makes $12/hour, her financial function is $m(t)=\text{12}t$ . We can graph it like this.

This simple graph has a great deal to tell us about Alice’s job, if we read it correctly.

• The graph contains the point $(\mathrm{3,}\text{300})$ .What does that tell us? That after Alice has worked for three hours, she has made $300.
• The graph goes through the origin (the point $(\mathrm{0,0})$ ). What does that tell us? That when she works 0 hours, Alice makes no money.
• The graph exists only in the first quadrant. What does that tell us? On the mathematical level, it indicates the domain of the function ( $t\ge 0$ ) and the range of the function ( $m\ge 0$ ). In terms of the situation, it tells us that Alice cannot work negative hours or make negative money.
• The graph is a straight line. What does that tell us? That Alice makes the same amount of money every day: every day, her money goes up by $100. ($100/day is the slope of the line—more on this in the section on linear functions.)

Consider now the following, more complicated graph, which represents Alice’s hair length as a function of time (where time is now measured in weeks instead of hours).

What does this graph $h(t)$ tell us? We can start with the same sort of simple analysis.

• The graph goes through the point $(\mathrm{0,}\text{12})$ .This tells us that at time $(t=0)$ , Alice’s hair is 12" long.
• The range of this graph appears to be . Alice never allows her hair to be shorter than 12" or longer than 18".