AP Precalculus Unit 1 Notes: Understanding Polynomial Functions

Rates of Change: Linear vs. Nonlinear Functions

What “rate of change” means

A rate of change describes how much the output of a function changes when the input changes. If a function is written as $y=f(x)$, then “rate of change over an interval” compares the change in $f(x)$ to the change in $x$.

The most common rate of change you’ll use in precalculus is the average rate of change from $x=a$ to $x=b$:

$\text{Average rate of change on }[a,b]=\frac{f(b)-f(a)}{b-a}$

This is the slope of the secant line through the points $\big(a,f(a)\big)$ and $\big(b,f(b)\big)$ on the graph.

Why it matters (big picture)

Rate of change is the bridge between:

• Algebra (computing differences and slopes)
• Graphs (steepness and direction)
• Modeling (interpreting real situations like speed, cost per item, or growth)

In Unit 1, it’s also a key way to distinguish linear behavior (constant rate) from polynomial behavior (rate typically changes).

Linear functions: constant rate of change

A linear function has the form:

$f(x)=mx+b$

• $m$ is the slope (constant rate of change).
• $b$ is the $y$-intercept.

If you compute the average rate of change on any interval for a linear function, you always get the same value $m$. That’s because the graph is a straight line—its steepness never changes.

Example 1 (constant average rate of change):
Let $f(x)=3x-2$. Compute the average rate of change from $x=1$ to $x=5$.

1. Compute outputs:
   • $f(1)=3(1)-2=1$
   • $f(5)=3(5)-2=13$
2. Apply the formula:

$\frac{f(5)-f(1)}{5-1}=\frac{13-1}{4}=3$

That equals the slope, as expected.

Nonlinear functions: changing rate of change

A function is nonlinear if its average rate of change depends on the interval you choose. For polynomials (like quadratics and cubics), the graph bends, so the steepness varies.

A common misconception is: “If the graph is increasing, the rate of change is constant.” Increasing only means outputs go up as inputs go up; it does not mean they go up at a constant pace.

Example 2 (average rate of change changes):
Let $f(x)=x^2$.

• From $x=1$ to $x=2$:

$\frac{f(2)-f(1)}{2-1}=\frac{4-1}{1}=3$

• From $x=2$ to $x=3$:

$\frac{f(3)-f(2)}{3-2}=\frac{9-4}{1}=5$

The average rate of change increased, showing the function is nonlinear.

Reading rate of change from representations

You may be given a graph, table, or equation.

• From a table, compute successive differences in $y$ compared to differences in $x$.
• From a graph, interpret slope of secant lines (average ROC) or overall “steepness changes.”
• From an equation, recognize linear vs polynomial degree: degree 1 is linear; degree 2 or higher is nonlinear.

Exam Focus

• Typical question patterns:
  • Compute and interpret average rate of change on a specified interval from an equation, table, or graph.
  • Decide whether a relationship is linear by checking whether average rate of change stays constant across intervals.
  • Compare rates of change on different intervals for a polynomial function and connect to “steepness” on the graph.
• Common mistakes:
  • Using $\frac{f(a)-f(b)}{b-a}$ (reversing one difference) and getting the wrong sign; keep the order consistent.
  • Assuming “increasing” implies “constant rate”; check multiple intervals.
  • Forgetting units in interpretation (for modeling): rate is “output units per input unit.”

Polynomials: Degree, Coefficients, and End Behavior

What a polynomial is

A polynomial function is a function that can be written as a sum of constant multiples of nonnegative integer powers of $x$:

$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0$

where:

• $n$ is a nonnegative integer.
• $a_n,\dots,a_0$ are real numbers called coefficients.
• $a_n\neq 0$ (otherwise the “true” highest power would be smaller).

Polynomials are central in precalculus because they are:

• Algebraically manageable (factoring, solving).
• Graphically rich (turning points, intercepts).
• Widely used in modeling (area, revenue, motion approximations).

Degree and leading coefficient

The degree of a polynomial is the highest exponent of $x$ with a nonzero coefficient. The **leading term** is $a_nx^n$, and $a_n$ is the leading coefficient.

Degree matters because it strongly predicts the “big picture” shape:

• The graph can have at most $n-1$ turning points.
• The end behavior is determined by the leading term.

A typical error is to misidentify degree from a factored form. For example, $f(x)=(x-2)^2(x+1)$ has degree $3$ because exponents add when you multiply factors.

Standard form vs factored form

You might see polynomials written in different ways:

End behavior (what happens as $x$ goes to $\infty$ or $-\infty$)

End behavior describes the left and right “tails” of the graph. For polynomials, end behavior is determined by the leading term $a_nx^n$.

Think of it this way: for very large $|x|$, the leading term dominates the rest of the polynomial, so the polynomial behaves like $a_nx^n$.

How degree parity and sign of leading coefficient control end behavior

1. Even degree (like $x^2$, $x^4$): both ends go in the same direction.

   • If $a_n>0$, both ends go up.
   • If $a_n<0$, both ends go down.

2. Odd degree (like $x^3$, $x^5$): ends go in opposite directions.

   • If $a_n>0$, left end down and right end up.
   • If $a_n<0$, left end up and right end down.

Example 1 (end behavior from leading term):
Determine the end behavior of:

$f(x)=-3x^5+2x^2-7$

• Degree: $5$ (odd)
• Leading coefficient: $-3$ (negative)

So the left end goes up and the right end goes down:

• As $x\to -\infty$, $f(x)\to \infty$.
• As $x\to \infty$, $f(x)\to -\infty$.

Example 2 (same degree, different leading coefficient):
Compare $x^4-100x$ and $-x^4-100x$. Both have degree $4$ (even), but the leading coefficient sign flips the end behavior (up-up vs down-down).

Connecting end behavior to graphs

On an exam, you may be shown multiple graphs and asked which matches a polynomial equation. End behavior is often the fastest way to eliminate wrong choices.

A common misconception is to let a large constant term (like $+5000$) “override” the end behavior. Constants can shift the graph up or down, but for large $|x|$ the leading term still dominates.

Exam Focus

• Typical question patterns:
  • Identify degree and leading coefficient from standard or factored form.
  • Match an equation to a graph using end behavior.
  • Predict end behavior from degree parity (even/odd) and sign of leading coefficient.
• Common mistakes:
  • Confusing degree with the number of terms (degree depends on exponents, not how many terms appear).
  • Using the constant term to decide end behavior; end behavior comes from the leading term.
  • Forgetting that multiplying factors adds degrees (exponents add in products).

Zeros and Multiplicity of Polynomial Functions

What zeros are

A zero (or root) of a polynomial is an $x$-value that makes the polynomial equal to zero:

$f(r)=0$

Graphically, zeros correspond to x-intercepts (where the graph crosses or touches the $x$-axis).

Zeros matter because they let you:

• Solve polynomial equations (find where the output is zero).
• Sketch graphs using intercepts and shape.
• Build polynomials from given intercept information (a common modeling and reasoning task).

Factored form and the meaning of a factor

If a polynomial is written in factored form, zeros are often visible immediately. The key idea is:

If $f(x)=(x-r)\cdot g(x)$, then $x=r$ makes $f(x)=0$.

More generally, a factor $ax+b$ corresponds to a zero at:

$x=-\frac{b}{a}$

Multiplicity: how “strongly” a zero occurs

The multiplicity of a zero tells you how many times a factor repeats. If $x=r$ is a zero with multiplicity $k$, then $f(x)$ contains the factor:

$(x-r)^k$

Multiplicity matters because it affects how the graph behaves at the x-intercept:

• Odd multiplicity: the graph crosses the x-axis at the zero (the sign of the function changes).
• Even multiplicity: the graph touches the x-axis and turns around (the sign does not change).

There’s also a “strength” intuition:

• Multiplicity $1$: usually crosses sharply.
• Multiplicity $2$: touches and bounces (like a parabola at its vertex).
• Multiplicity $3$: crosses but flattens near the axis.

A common mistake is to think “touching” always means the graph has a local minimum or maximum there. It does for even multiplicity, but the overall context still matters; it’s safer to say “touches and turns” rather than name the extremum.

Worked example: zeros and multiplicity from an equation

Consider:

$f(x)=(x+2)^2(x-1)^3$

1. Zeros come from setting each factor to zero:
   • $x+2=0$ gives $x=-2$
   • $x-1=0$ gives $x=1$
2. Multiplicities:
   • $x=-2$ has multiplicity $2$ (even) so the graph touches and turns.
   • $x=1$ has multiplicity $3$ (odd) so the graph crosses, likely flattening near the intercept.
3. Degree check: total degree is $2+3=5$, so end behavior should match an odd-degree polynomial.

This “degree check” is an excellent habit: it helps catch algebra slips like forgetting an exponent.

Building a polynomial from zeros (and one additional point)

Often you’re told intercepts and asked to write a polynomial. If the zeros are $r_1,r_2,\dots$ with multiplicities, then a general form is:

$f(x)=a(x-r_1)^{k_1}(x-r_2)^{k_2}\cdots$

where $a$ is a nonzero constant controlling vertical stretch and possibly reflection.

To determine $a$, you need one more piece of information—commonly a point on the graph like a y-intercept.

Example: write a polynomial with given zeros and a point
Zeros: $x=-1$ (multiplicity 2), $x=3$ (multiplicity 1). Point: $f(0)=18$.

1. Start with the factored form:

$f(x)=a(x+1)^2(x-3)$

2. Use the point $x=0$:

$18=f(0)=a(0+1)^2(0-3)=a(1)(-3)=-3a$

3. Solve for $a$:

$a=-6$

So:

$f(x)=-6(x+1)^2(x-3)$

The Fundamental Theorem of Algebra (what you need here)

A key fact used in polynomial reasoning is: a polynomial of degree $n$ has exactly $n$ complex zeros counting multiplicity. In AP Precalculus, this supports reasoning like “a degree 5 polynomial can’t have 6 real x-intercepts” and “if you already have accounted for multiplicities totaling 4, one more zero remains (possibly non-real).”

You typically do not need to find complex zeros in this section, but you do need the counting idea for interpreting graphs and factor structures.

Exam Focus

• Typical question patterns:
  • Given a factored polynomial, identify zeros and state how the graph behaves at each x-intercept (cross vs touch).
  • Given a graph with intercept behavior, write a possible factored form including multiplicities.
  • Find the unknown leading factor $a$ using an additional point such as $f(0)$.
• Common mistakes:
  • Mixing up the sign inside a factor: zero at $x=r$ corresponds to factor $(x-r)$, not $(x+r)$.
  • Saying “even multiplicity crosses” (it does not); even multiplicity touches/turns.
  • Forgetting to include the multiplier $a$ when constructing a polynomial from zeros; without $a$ you have only a family of functions.

Polynomial Inequalities and Sign Analysis

What a polynomial inequality is

A polynomial inequality asks where a polynomial expression is positive, negative, or zero, such as:

$f(x)>0$

$f(x)\ge 0$

$f(x)<0$

$f(x)\le 0$

These problems are about understanding the sign (positive/negative) of the polynomial across different intervals of $x$.

Why sign analysis works

Polynomials are continuous functions. That means they can only change sign by passing through zero—so the only places where the sign might change are at the zeros.

This is the key strategy:

1. Factor the polynomial (if possible).
2. Find its zeros (critical points).
3. Use a sign chart (test intervals) to determine where the expression is positive or negative.
4. Include or exclude zeros depending on whether the inequality is strict (>, <) or inclusive ($\ge,\le$).

A common mistake is trying to “solve inequalities like equations” by setting each factor separately greater than zero without tracking interactions between factors. The product’s sign depends on all factors together.

Sign changes and multiplicity (an important connection)

Multiplicity from the previous section becomes very useful here.

• At a zero with odd multiplicity, the polynomial’s sign changes as you pass through the zero.
• At a zero with even multiplicity, the polynomial’s sign stays the same (it touches 0 and returns).

This can reduce the amount of “testing” you need in a sign chart, and it helps you reason from a graph.

Worked inequality example 1: factoring and sign chart

Solve:

$x^3-4x^2-x+4>0$

Step 1: Factor by grouping.
Group terms:

$x^3-4x^2-x+4=x^2(x-4)-1(x-4)$

Factor out the common binomial:

$x^3-4x^2-x+4=(x-4)(x^2-1)$

Then factor the difference of squares:

$x^2-1=(x-1)(x+1)$

So:

$x^3-4x^2-x+4=(x-4)(x-1)(x+1)$

Step 2: Find critical points (zeros).
Zeros are $x=-1,1,4$.

Step 3: Create intervals and test signs.
Intervals:

• $(-\infty,-1)$
• $(-1,1)$
• $(1,4)$
• $(4,\infty)$

Pick a test point in each interval and evaluate the sign of the product.

• For $x=-2$:
  • $(x-4)<0$, $(x-1)<0$, $(x+1)<0$
  • Product of three negatives is negative, so expression < 0.
• For $x=0$:
  • $(x-4)<0$, $(x-1)<0$, $(x+1)>0$
  • Two negatives and one positive gives positive.
• For $x=2$:
  • $(x-4)<0$, $(x-1)>0$, $(x+1)>0$
  • One negative and two positives gives negative.
• For $x=5$:
  • All three factors positive, so product positive.

So the inequality $>0$ holds on:

$(-1,1)\cup(4,\infty)$

Because it’s a strict inequality, you do not include $-1,1,4$.

Worked inequality example 2: repeated factor (even multiplicity)

Solve:

$(x-2)^2(x+3)\le 0$

Step 1: Identify zeros and multiplicities.

• $x=2$ has multiplicity 2 (even).
• $x=-3$ has multiplicity 1 (odd).

Step 2: Determine sign behavior by intervals.
Critical points split the number line into:

• $(-\infty,-3)$
• $(-3,2)$
• $(2,\infty)$

Now reason about signs:

• The factor $(x-2)^2$ is always $\ge 0$, and it is 0 only at $x=2$.
• So the sign of the whole product (away from $x=2$) is controlled by $(x+3)$.

Check intervals:

• If $x<-3$, then $(x+3)<0$, and $(x-2)^2>0$, so product < 0.
• If $x>-3$ and $x\neq 2$, then $(x+3)>0$, and $(x-2)^2>0$, so product > 0.

Because the inequality is $\le 0$, include where the product is negative and where it is zero:

• Negative on $(-\infty,-3)$
• Zero at $x=-3$ and $x=2$

Solution:

$(-\infty,-3]\cup\{2\}$

Notice how even multiplicity at $x=2$ means the sign does not flip there; it just touches 0.

Connecting sign analysis to graphs

If you’re given a graph of a polynomial and asked where $f(x)>0$ or $f(x)<0$:

• $f(x)>0$ corresponds to where the graph is above the $x$-axis.
• $f(x)<0$ corresponds to where it’s below.
• Zeros are x-intercepts; whether the graph crosses or touches helps you infer multiplicity and sign changes.

A frequent mistake is mixing up $x$-intervals (input values) with $y$-values. Inequalities like $f(x)>0$ ask about which $x$ make the output positive.

Exam Focus

• Typical question patterns:
  • Solve polynomial inequalities by factoring and using a sign chart.
  • Use a graph to determine solution intervals to $f(x)\ge 0$ or $f(x)<0$.
  • Use multiplicity information to predict whether the sign changes at each intercept.
• Common mistakes:
  • Forgetting to include zeros when the inequality is $\ge 0$ or $\le 0$.
  • Testing points incorrectly because intervals were set up wrong; always list critical points in order and build intervals between them.
  • Assuming the sign always flips at every zero; it flips only at zeros of odd multiplicity.