How do you evaluate expressions step by step?
Substitute, then calculate in order. Write each substitution in parentheses so signs and powers apply to the whole value.
- Rewrite the expression with the number in place of the variable, inside parentheses. Copy every other part exactly as it was.
- Work out any powers.
- Do multiplication and division from left to right.
- Finish with addition and subtraction from left to right.
Worked example: 2x² + 3x − 1 at x = 4
The value is 43. Square first, then multiply, then add and subtract.
In the calculator, type this as 2x^2 + 3x - 1 @ x = 4. The @ sign separates the expression from the value.
Why do you need parentheses when you plug in a negative number?
Without parentheses, the power applies only to the digit and you lose the negative sign. (−3)² is 9, but −3² is −9.
Take x³ − 2x at x = −3. With parentheses it reads (−3)³ − 2(−3). The cube is −27 and −2 × −3 is +6, so the result is −27 + 6 = −21.
One more to watch: x² + 4x + 4 at x = −5 is 25 − 20 + 4 = 9. An even power of a negative is positive; an odd power stays negative.
What order of operations do you use when evaluating?
Use PEMDAS: parentheses, exponents, multiplication and division, then addition and subtraction. Multiplication and division share a level, as do addition and subtraction, and you go left to right within each.
| Expression | Value | Why |
|---|---|---|
| 3x² at x = −2 | 12 | Square first: 3 × 4 |
| −x² at x = 3 | −9 | Square 3, then apply the minus |
| 3(x + 4) at x = 2 | 18 | Bracket first: 3 × 6 |
| x² − 3x + 2 at x = −1 | 6 | 1 + 3 + 2 |
The second row trips up many students. In −x², the power belongs to x only, so you square 3 and then make it negative.
Brackets in the expression change the order. In 3(x + 4) you add first because the bracket comes first, then multiply. If you multiplied 3 by x alone, you'd get 3(2) + 4 = 10 instead of 18.
How do you evaluate an expression with a fraction?
Substitute the fraction in parentheses and use fraction arithmetic. For 5a − 7 at a = 1/2, you get 5(1/2) − 7 = 5/2 − 14/2 = −9/2.
The calculator keeps fractions exact, so it returns −9/2 and not −4.5. Type it as 5a - 7 @ a = 1/2. Decimals work as well: 4x − 2 at x = 0.5 gives 0.
A fraction bar in the expression itself is a division. For x/2 + 3 at x = 4, divide first to get 2, then add 3 for a final value of 5.
What does f(4) mean, and how do you find it?
f(4) means the value of the function f when x is 4. You find it by evaluating the rule for f at x = 4, which is exactly what this page does.
Read f(4) as "f of 4". It does not mean f times 4. The letter f is the name of the rule, and the number in the parentheses is the input you plug in.
If f(x) = x² − 5x + 6, then f(4) = 16 − 20 + 6 = 2. To get this from the calculator, type only the right side with the value: x^2 - 5x + 6 @ x = 4.
How do you make a table of values by evaluating?
Evaluate the same expression at several x values and list the results. That table is what you plot when you graph by hand.
Here is x² − 5x + 6 for x from 0 to 4. Each row is one evaluation.
| x | x² − 5x + 6 |
|---|---|
| x = 0 | 6 |
| x = 1 | 2 |
| x = 2 | 0 |
| x = 3 | 0 |
| x = 4 | 2 |
The two zeros at x = 2 and x = 3 are where the graph crosses the x-axis. They are also the solutions of x² − 5x + 6 = 0, so evaluating can confirm the zeros a solver gives you.
What mistakes should you avoid when plugging in values?
Leaving out the parentheses around a negative is the big one. Another is multiplying before squaring: 2x² at x = 4 is 2 × 16 = 32, not 8² = 64.
Also watch implied multiplication. 3x at x = 4 is 3 × 4 = 12, not 34. Writing 3(4) instead of 34 keeps that straight.