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Evaluation: Evaluating Expressions,
     Polynomials, and Functions
(page 2 of 2)

Sections: Evaluating Expressions and Polynomials, Evaluating Functions

  • Evaluate y = 4x 3 at x = 0.
    • y = 4(0) 3 = 0 3 = 3

    This means that the point (0, 3) is on the line.

  • Evaluate y = 4x 3 at x = 3.
    • y = 4(3) 3 = 12 3 = 9

    Then (3, 9) is on the line.


By the way, evaluating the same equation at three or more points like this, and getting a list of points, is how you plot points and graph equations.

In this case, the points from the evaluating (including the point from the previous page) are: (1, 7), (0, 3), and (3, 9).

Then the graph looks like the drawing at the right:

Copyright Elizabeth Stapel 2000-2011 All Rights Reserved

You can verify from the picture that the three points I found are indeed on the graph.


y = 4x - 3

You will also need eventually to evaluate functions.

    For the following exercises, let f(x) = sqrt(25 - x^2).

  • Evaluate f(3).

    To evaluate a function, I do just what I did above: I plug in the given value for x. Here, I am supposed to evaluate at the value x = 3. The notation is different, but "f(3)" means exactly the same thing as "evaluate f(x) = sqrt(25 - x^2) at x = 3"!

      f(-3) = 4

Note how I used parentheses. It is very easy to mess up the minus signs if you're not careful and use lots of parentheses. Take the time to be careful!


  • Evaluate f(3).
    • f(3) = 4

  • Evaluate f(1).
    • f(-1) = sqrt(24)

Note that I gave the answer in two formats: the "exact" form (with the radical in it) and the "approximate" form (with the wiggly "equals"). Usually you will be expected to evaluate exactly; that is, it will usually be correct to leave the answer in a messy form (with a radical, or a fraction, or with pi in it [instead of rounding to 3.14], etc). However, there are times when the approximate form is better. Often, word problems need an answer that can be applied in "real life". For instance, "square root of 24 meters" isn't very useful when you're trying to figure out to what length to cut a board, but "about 4.9 meters" is perfectly useful, and probably quite accurate enough for whatever you're building. You will also need to approximate for when you're graphing. For instance, I would have no idea where to plot the square root of 24, but I know right where to draw the line for 4.9.

By the way, you graph functions just like you graph other equations: by evaluating the function at a few values of x, drawing the points, and connecting the dots. (This is exactly what a graphing calculator does, by the way.) The graph of the function used in the three examples above looks like this:

f(x) = sqrt(25 - x^2)

Just remember: "evaluate" means "plug-n-chug". Be careful with the subtractions, negatives, and exponents (by using parentheses appropriately). Don't try to do too much at once; don't skip steps, don't try to do three steps at once, and don't try to do everything in your head. Take your time, and evaluation problems should work out fine!

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Cite this article as:

Stapel, Elizabeth. "Evaluation: Functions." Purplemath. Available from Accessed


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