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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 some 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 you plot a few other points on your T-chart, it will be no trouble to graph this:

T-chart	Graph (Note scale on axes!)

(For further information, please study the lesson on "Graphing Polynomial Functions".)

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 you know that 2x – 5 must not be negative. Algebraically, you must have 2x – 5 > 0. If you solve this inequality, you will come up with the domain for the function y being x > 2.5. So, for heaven's sake, please don't try to plot points that aren't allowed!   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

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

Okay T-chart	Incorrect graph

By not paying attention to the domain and by not plotted negative x-values, the student fooled himself into thinking that this graphed as a straight line. Instead, you want to do the graph like this:

Better T-chart	Correct graph

Note that radical-function graphs are generally curvey like this; they are not straight lines. (For further information, please study the lesson on "Graphing Radical Functions".)

Rational functions

Before you even draw a T-chart for a rational function, you first have to find the asymptotes and intercepts. Once you have successfully done that, you can then choose x's between the x-intercepts and vertical asymptotes, to give you 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 you've learned about rational functions, you 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. Now plot a few additional 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.) (For further information, please study the lesson on "Graphing Rational Functions".)

Piecewise functions

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

Since this has two pieces, you may find it helpful to do two T-charts; if it had more pieces, you could do more T-charts. The break between the two "halves" of the function (the point at which the function changes rules) is at x = 1, so that is where your 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? 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, I 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.

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