Notes
Rates of change
A function's average rate of change on $[a, b]$ is the slope of the secant line:
- Increasing means outputs rise as inputs rise. Concave up means the rate of change is increasing (the graph bends up like a cup).
- Over equal-length input intervals, a linear function has constant rates of change and a quadratic has rates of change that change at a constant rate.
Polynomial functions
A polynomial has the form $p(x) = a_nx^n + \dots + a_1x + a_0$. The degree $n$ and the leading coefficient $a_n$ control the end behavior.
| Degree | Leading coefficient | Left end | Right end |
|---|---|---|---|
| Even | Positive | Up | Up |
| Even | Negative | Down | Down |
| Odd | Positive | Down | Up |
| Odd | Negative | Up | Down |
- A real zero $x = a$ means $(x - a)$ is a factor. If the factor appears an even number of times, the graph touches the axis and turns around; odd, it crosses.
- A degree $n$ polynomial has at most $n$ real zeros and at most $n - 1$ turning points. Non-real zeros come in conjugate pairs $a \pm bi$.
- Even functions satisfy $f(-x) = f(x)$ (symmetric over the $y$-axis). Odd functions satisfy $f(-x) = -f(x)$ (symmetric about the origin).
Rational functions
A rational function is a ratio of polynomials, $\displaystyle r(x) = \frac{p(x)}{q(x)}$.
- End behavior comes from the leading terms. If the degrees are equal, the horizontal asymptote is the ratio of leading coefficients. If the bottom has the higher degree, $y = 0$. If the top is exactly one degree higher, there is a slant asymptote.
- A zero of the denominator that does not cancel gives a vertical asymptote. A factor that cancels from top and bottom gives a hole.
- Zeros of the rational function come from zeros of the numerator that don't cancel.
Transformations and modeling
$g(x) = a\,f(b(x - h)) + k$ stretches vertically by $a$, compresses horizontally by $b$, and shifts right $h$ and up $k$.
- Choose a model by the data's pattern: constant differences suggest linear, constant second differences suggest quadratic.
Topics in this unit
Each topic has its own short notes and practice.
- 1.1Change in Tandem
- 1.2Rates of Change
- 1.3Rates of Change in Linear and Quadratic Functions
- 1.4Polynomial Functions and Rates of Change
- 1.5Polynomial Functions and Complex Zeros
- 1.6Polynomial Functions and End Behavior
- 1.7Rational Functions and End Behavior
- 1.8Rational Functions and Zeros
- 1.9Rational Functions and Vertical Asymptotes
- 1.10Rational Functions and Holes
- 1.11Equivalent Representations of Polynomial and Rational Expressions
- 1.12Transformations of Functions
- 1.13Function Model Selection and Assumption Articulation
- 1.14Function Model Construction and Application
Practice
$f(4) = 4$ and $f(1) = -2$, so $\displaystyle \frac{4 - (-2)}{4 - 1} = \frac{6}{3} = 2$.
Odd degree with a negative leading coefficient rises on the left and falls on the right.
The factor $(x - 2)$ appears twice (even multiplicity), so the graph touches and turns.
The degrees are equal, so the asymptote is the ratio of leading coefficients: $\displaystyle \frac{6}{3} = 2$.
Both factor to share $(x - 3)$: $\displaystyle \frac{(x-3)(x+3)}{(x-3)(x+2)}$. The common factor cancels, leaving a hole at $x = 3$.
Non-real zeros come in conjugate pairs, so $2 - i$ is also a zero. That's three zeros, so the degree is at least 3.
$f(-x) = -x^3 + 5x = -(x^3 - 5x) = -f(x)$. The "+1" in $x^3 + 1$ breaks odd symmetry.
$x - 4$ inside shifts right 4; $+2$ outside shifts up 2.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A model for the height of a plant, in centimeters, $t$ weeks after planting is $h(t) = \frac{30t}{t + 5}$ for $t \ge 0$.
(a) Find the average rate of change of $h$ from $t = 0$ to $t = 5$. Give units.
Show the answer and scoring guide
$\displaystyle h(5) = \frac{150}{10} = 15$ and $h(0) = 0$, so $\displaystyle \frac{15 - 0}{5 - 0} = 3$ centimeters per week.
Check off the points you earned:
(b) Find $\lim_{t \to \infty} h(t)$ and explain what it means in context.
Show the answer and scoring guide
The degrees are equal, so the limit is $\displaystyle \frac{30}{1} = 30$. The plant's height levels off near 30 centimeters as time goes on.
Check off the points you earned:
(c) Is $h$ concave up or concave down for $t > 0$? Explain using rates of change.
Show the answer and scoring guide
Concave down. Over equal intervals the growth gets smaller (for example, 15 cm in weeks 0 to 5 but only 5 cm in weeks 5 to 10, since $h(10) = 20$), so the rate of change is decreasing.
Check off the points you earned: