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

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

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)