Print-friendly page

Graphing Overview (page 3 of 3)

Sections: Straight lines, Absolute values & quadratics, Polynomials, radicals, rationals, & piecewise

General polynomials

For graphing higher-power polynomials, you will of course need plenty of points. The best points are the intercepts; to find these, use all the factoring tools that you have. Then also pick x's between the x-intercepts, and plot. If you keep in mind the end-behavior of polynomials, then these graphs can actually be not too hard to do. For example, let y = x⁴ – 13x² + 36. This is a positive even power ("to the fourth"), so the graph will be up on both ends (like the quadratic above). Factoring the polynomial, we get y = (x + 3)(x – 3)(x + 2)(x – 2), so the zeroes (x-intercepts) are –3, –2, 2, and 3. If we plot a few other points on our T-chart, it will be no trouble to graph this:

T-chart	Graph (Note scale on axes!)

Radical functions

The most important thing to remember here is that, if you're dealing with a square root, you cannot have a negative inside the radical. Since this is true, it is entirely possible, even likely, that there will be values of x that are not allowed inside the function. For instance, if y = sqrt(2x – 5), then we know that 2x – 5 must not be negative. Algebraically, we must have 2x – 5 > 0. If we solve this, we come up with the domain for y being x > 2.5. So, for heaven's sake, please don't try to plot points that aren't allowed!

The following is often what happens, if the student is careless:

Okay T-chart	Incorrect graph

Instead, you want to do the graph like this:   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

Better T-chart	Correct graph

Note that radical-function graphs are generally curvey like this; they are not straight lines!

Rational functions

Before we even draw a T-chart for a rational function, we first have to find the asymptotes and intercepts. Once we have successfully done that, we can then choose x's between the x-intercepts and vertical asymptotes, to give us the additional information necessary to graph the function. Actually, as bad as these functions look, they are quite easy to graph. For instance, suppose you have:

From what we know about rational functions, we know that the vertical asymptotes are at x = 2 and x = –2, the horizontal asymptote is at y = 2, the x-intercepts are at x = –3 and x = 3, and the y-intercept is at y = 4.5. We'll plot a few points between these other points:

T-chart	Graph

(Remember that horizontal asymptotes are just "suggestions" off to the sides; they mean next to nothing in the "middle", and you're quite welcome to cross them.)

How did I know which way to go at the vertical asymptotes? Go back and look at the x-intercepts we had. We can only cross the x-axis at an intercept; therefore, if there is no intercept, then there is no crossing of the axis. So, on the left, we knew the graph traced along the horizontal asymptote, came down to cross at x = –3, and then stayed down, because there was no place to cross to get back up. In the middle, there were no x-intercepts, but there were points above the x-axis, so the graph was always above. On the right, the graph works the same as it did on the left. (Occasionally the graph just touches the x-axis at an intercept, instead of going though.That's why we checked points between the x-intercepts and the vertical asymptotes.)

Piecewise functions

Since piecewise functions are defined in pieces, then we have to graph them in pieces, too. For instance, suppose we have:

Since this has two pieces, we will do two T-charts; if it had more pieces, we would do more T-charts. The break is at x = 1, so that is where our T-charts will break. The procedure looks like this:

T-chart 1	T-chart 2	Graph

Why did I list that last point in T-chart 1 in parentheses? Because, technically speaking, x = 1 does not belong on that chart. But it is often helpful to know where the function "almost" is at the end. That's why, on the graph, we drew that point as an open circle, meaning that the graph is everything up to, but not including, that point. This can be especially important when, as in this case, the pieces of the function don't join up at the ends.

<< Previous  Top  |  1 | 2 | 3  |  Return to Index

  Copyright © 2006-2008  Elizabeth Stapel   |   About   |   Terms of Use

 Feedback   |   Error?