Graphing: Intercepts, End Behavior, and the Line It Never Reaches

In this lesson you will: Graph y=a* b^x from a short table of integer inputs; Read the y-intercept off the rule, and explain why there is no x-intercept; Describe end behavior in both directions and draw the horizontal asymptote correctly.

Pick inputs you can do in your head

Four or five points are enough to show the shape, provided you take some negative inputs. Students who plot only x=0,1,2,3 get a curve that looks like it starts at the y-axis, which hides half the story.

For y=3* 2^x: at x=-2 you get 3*1/4=0.75; at x=-1, 1.5; at x=0, 3; then 6, 12, 24. The left side flattens, the right side takes off.

The line it gets close to and never reaches

Every value of 3* 2^x is a positive number times a positive number, so every output is positive. The graph therefore stays strictly above the x-axis, no matter how far left you go.

At x=-10 the output is 3* 2^-10=(3)/(1024), which is about 0.0029. Small. Not zero. At x=-20 it is smaller still, and still not zero. The x-axis is a horizontal asymptote: the curve gets arbitrarily close to it and never touches it.

That is the same statement as \"this function has no x-intercept.\" Solving 3* 2^x=0 would require 2^x=0, and no power of 2 is zero.

End behavior, read off the base

End behavior is the answer to two questions: what happens far to the right, and what happens far to the left. For an exponential the base answers both at once.

Notice the symmetry. Growth rises to the right and flattens to the left; decay does exactly the reverse. The asymptote is y=0 either way — it is only the end it belongs to that changes.

Decay, plotted

Here is y=16(1/2)^x. At x=-1 it is 32; at x=0, 16; then 8, 4, 2, 1.

Halving repeatedly makes a number small quickly, but it never makes it zero. Halving 16 a hundred times leaves a number with about thirty zeros after the decimal point — and a nonzero digit at the end of them.

Common mistake — Drawing the curve onto the x-axis

A student sketches y=5* 2^x, and on the left the curve settles down onto the x-axis and runs along it like a runway.

It is an honest drawing problem before it is a mathematical one. By x=-4 the output is 0.3125, and at the scale of a classroom graph that is less than a pencil width above the axis. The hand does the rest.

The consequence is not cosmetic. A graph touching the axis claims the function has an x-intercept, so it claims 5* 2^x=0 has a solution. It does not. Two positive numbers multiplied never produce zero.

How to catch it: Pick the leftmost x on your sketch, say x=-4, and evaluate: 5* 2^-4=(5)/(16)=0.3125. Now find 0.3125 on your y-axis. If your curve is drawn lower than that, it is wrong — and it will be wrong in the same way for every x further left.