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Evaluation: Evaluating Expressions,
     Polynomials, and Functions

"Evaluation" mostly means "simplifying an expression down to a single numerical value". Sometimes you will be given a numerical expression, where all you have to do is simplify; that is more of an order-of-operations kind of question. In this lesson, I'll concentrate on the "plug and chug" aspect of evaluation: plugging in values for variables, and "chugging" my way to the simplified answer.

Usually the only hard part in evaluation is in keeping track of the minus signs. I would strongly recommend that you use parentheses liberally, especially when you're just getting started.

• Evaluate a²b for a = –2, b = 3, c = –4, and d = 4.

To find my answer, I just plug in the given values, being careful to use parentheses, particularly around the minus signs:   Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

(–2)²(3) = (4)(3) = 12

• Evaluate a – cd for a = –2, b = 3, c = –4, and d = 4.

(–2) – (–4)(4) = –2 – (–16) = –2 + 16 = 16 – 2 = 14

• Evaluate (b + d)² for a = –2, b = 3, c = –4, and d = 4.

I must take care not to try to "distribute" the exponent through the parentheses. Exponents do NOT distribute over addition! I should never try to say that (b + d)² is the same as b² + d²! They are NOT the same thing! I must evaluate the expression as it stands:

( (3) + (4) )² = ( 7 )² = 49

• Evaluate b² + d² for a = –2, b = 3, c = –4, and d = 4.

(3)² + (4)² = 9 + 16 = 25

Notice that this does not match the answer to the previous evaluation, pointing out again that exponents do not "distribute" the way multiplication does.

• Evaluate bc³ – ad for a = –2, b = 3, c = –4, and d = 4.

(3)(–4)³ – (–2)(4) = (3)(–64) – (–8) = –192 + 8 = –184

The most common "expression" you'll likely need to evaluate will be polynomials. To evaluate, you take the polynomial and plug in a value for x.

• Evaluate x⁴ + 3x³ – x² + 6 for x = –3.

(–3)⁴ + 3(–3)³ – (–3)² + 6
    = 81 + 3(–27) – (9) + 6
    = 81 – 81 – 9 + 6
    = –3

• Evaluate 3x² – 12x + 4 for x = –2.

3(–2)² – 12(–2) + 4 = 3(4) + 24 + 4 = 12 + 24 + 4 = 40

• Evaluate y = 4x – 3 at x = –1.

y = 4(–1) – 3 = –4 – 3 = –7

Note: This means that the point (–1, –7) is on the line y = 4x – 3.

• 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.

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

You will also need eventually to evaluate functions.

For the following exercises, let .

• 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 at x = –3"!

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).

• Evaluate f(–1).

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:

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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