Polynomial Functions

Rate of change

A function's rate of change between two points is the same slope formula used for lines: the change in output divided by the change in input.

$$\text{rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$$

For a linear function this rate is constant everywhere. For a quadratic or higher-degree function, it describes only the average rate between those two specific points. Example: a function passes through $(2, 5)$ and $(6, 17)$, so its rate of change is $\dfrac{17-5}{6-2} = \dfrac{12}{4} = 3$.

Keep the subtraction order consistent. If you subtract the outputs as (second minus first), do the inputs the same way. Reversing only one of them flips the sign. Also remember to finish the division: stopping at $6 - 2 = 4$ or at $12$ is a frequent slip.

When the points belong to a line, you can go further. Once you know the slope, plug one point into $y = mx + b$ to find the y-intercept. A line through $(2, 11)$ and $(5, 20)$ has slope $\dfrac{20-11}{5-2} = 3$, so $11 = 3(2) + b$ and $b = 5$. The y-intercept is $5$, not $11$. The $y$-coordinate of a given point is not the y-intercept unless the point sits at $x = 0$.

Zeros and factoring

A zero (or x-intercept) of a function is an input that makes the output $0$. To find the zeros of a quadratic, set it equal to zero and factor. For $f(x) = x^2 + x - 12$, look for two numbers that multiply to $-12$ and add to $1$: those are $4$ and $-3$, so

$(x + 4)(x - 3) = 0$, which gives $x = -4$ or $x = 3$.

The signs in the factors are the opposite of the signs of the zeros. That is the most common error here, and one that the answer choices are written to catch. A quick check is always available: substitute your zeros back into the original function.

Two special patterns appear often. A difference of squares factors as $x^2 - 9 = (x-3)(x+3)$, so the zeros are $\pm 3$, not $9$. For higher-degree polynomials, factor out the greatest common factor first: $x^3 - 4x = x(x^2 - 4) = x(x-2)(x+2)$, which has three real zeros: $0$, $2$, and $-2$. Do not lose the zero that comes from the factor $x$ itself.

End behavior

The end behavior of a polynomial is controlled entirely by its degree and leading coefficient.

• Even degree: both ends point in the same direction, up if the leading coefficient is positive and down if it is negative.
• Odd degree: the ends point in opposite directions. With a positive leading coefficient the graph rises on the right and falls on the left; with a negative leading coefficient it is the reverse.

End behavior from degree and leading coefficient

Worked example. Describe the end behavior of $f(x) = 3x^5 - 2x^2 + 1$.

Degree 5 is odd; the leading coefficient is positive.

Odd degree: the ends point in opposite directions.

Positive leading coefficient: up on the right, down on the left.

Only the term with the highest power matters. Ignore the other terms. If the polynomial is written in factored form, such as $(1-x)(x+2)(x^2+1)$, find the leading term by multiplying the highest-power term from each factor: $(-x)(x)(x^2) = -x^4$. Degree 4, negative leading coefficient: both ends point down.

Arithmetic sequences

The next section previews one arithmetic-sequence skill early, since it uses the same algebraic habit as the polynomial work just covered: setting up and solving for an unknown from a general formula.

An arithmetic sequence changes by the same constant difference $d$ at every step:

$$a_n = a_1 + (n-1)d.$$

Finding a term far down the sequence, or finding the common difference from two known terms that are not adjacent, both use this same formula, just solved for a different piece of it. The most frequent error is an off-by-one: to get from $a_1$ to $a_{12}$ you add $d$ eleven times, not twelve.

Arithmetic sequences from non-adjacent terms

Worked example. An arithmetic sequence has $a_2 = 9$ and $a_5 = 21$. Find $a_{10}$.

Find the common difference first, then extend to the requested term.

From $a_2$ to $a_5$ is 3 steps: $d = \dfrac{21-9}{3} = 4$.

$a_1 = 9 - 4 = 5$.

$a_{10} = 5 + (10-1)(4) = 41$.

Another route is to skip $a_1$ entirely: $a_{10}$ is 5 steps past $a_5$, so $a_{10} = 21 + 5(4) = 41$. Counting steps between known terms is often quicker than going back to the first term.

Where this appears as a model

Polynomials describe quantities built from products of lengths. Suppose you cut a square of side $x$ from each corner of a 12-by-8 sheet and fold up the sides. The box has dimensions $x$ by $(12-2x)$ by $(8-2x)$, so its volume is $x(12-2x)(8-2x)$. With $x = 2$, the volume is $2(12-4)(8-4) = 64$ cubic inches. The same expression works for any cut size $x$.

Strategy summary

• Rate of change is "change in output over change in input." Subtract in the same order top and bottom, and finish the division.
• Factor to find zeros, then flip the signs of the constants in the factors. Take out common factors first.
• For end behavior, look only at the degree (even or odd) and the sign of the leading coefficient.
• For sequences, count the steps between terms. From $a_k$ to $a_n$ is $n-k$ steps.