Unit 1: Polynomial and Rational Functions

21 terms. Say what each one means, then open the card to check.

• Average rate of change

  The change in output divided by the change in input over an interval, $\frac{f(b)-f(a)}{b-a}$. It is the slope of the line through the two endpoints.

  Example. If $f(1)=3$ and $f(5)=11$, the average rate of change on $[1,5]$ is $\frac{11-3}{5-1}=2$.

• Increasing on an interval

  A function is increasing on an interval when its output gets larger as its input gets larger, for every pair of inputs in that interval.

  Example. A tank that fills steadily from 8 a.m. to 9 a.m. has a volume function that is increasing on that hour.

• Concave up

  A graph is concave up on an interval when its rate of change is increasing there. The graph bends upward, like part of a cup.

  Example. Values 3, 4, 7, 12 at equally spaced inputs rise by 1, 3, then 5, so the rates grow and the graph looks concave up.

• Concave down

  A graph is concave down on an interval when its rate of change is decreasing there. The graph bends downward, like part of a dome.

  Example. A ball thrown upward has a height graph that is concave down: it rises more slowly, then falls faster and faster.

• Point of inflection

  A point on a graph where the rate of change switches from increasing to decreasing, or the reverse, so the concavity changes.

  Example. If the rates over equal steps are 4, -4, -6, -2 and 8, they shrink and then grow, so the concavity switches from down to up around the third step.

• Degree of a polynomial

  The greatest exponent of the variable in a polynomial written in standard form. It limits how many real zeros and turning points the graph can have.

  Example. $p(x)=4x^3-x+9$ has degree 3, so it has at most 3 real zeros and at most 2 local extrema.

• Zero of a function

  An input $a$ with $f(a)=0$. For a polynomial, $x-a$ is then a factor, and a real zero is an $x$-intercept of the graph.

  Example. For $p(x)=(x-4)(x+1)$, the zeros are 4 and -1, and the graph crosses the $x$-axis at both.

• Multiplicity

  The number of times a factor $x-a$ appears in a polynomial. At an even multiplicity the graph touches the axis and turns back; at an odd one it crosses.

  Example. In $(x-3)^2(x+2)$, the zero 3 has multiplicity 2, so the graph touches the $x$-axis at $x=3$ without crossing.

• Local maximum or minimum

  A point where a function switches between increasing and decreasing. A local maximum is higher than nearby points; a local minimum is lower.

  Example. If $f$ rises on $[0,2]$ and falls on $[2,5]$, then $f$ has a local maximum at $x=2$.

• Intermediate Value Theorem

  If $f$ is continuous on $[a,b]$, then $f$ takes every value between $f(a)$ and $f(b)$ somewhere on that interval.

  Example. A polynomial with $q(1)=-3$ and $q(4)=2$ must equal 0 at some $x$ between 1 and 4, since 0 lies between -3 and 2.

• End behavior

  What the outputs of a function do as the input increases or decreases without bound.

  Example. For $f(x)=x^2$, $f(x)$ grows without bound as $x$ goes far to the right and as $x$ goes far to the left.

• Leading term

  The term with the greatest exponent in a polynomial. For large inputs it outweighs every other term, so it sets the end behavior.

  Example. In $-2x^5+40x^2$, the leading term is $-2x^5$, so the outputs fall without bound as $x$ grows large.

• Vertical asymptote

  A vertical line $x=a$ that a graph approaches as its outputs grow without bound. For a rational function it comes from a zero of the denominator that does not cancel.

  Example. $r(x)=\frac{1}{x-4}$ has a vertical asymptote at $x=4$, because the outputs blow up as $x$ nears 4.

• Horizontal asymptote

  A horizontal line $y=c$ that the outputs approach as the input grows without bound in either direction.

  Example. $\frac{6x^2+1}{2x^2-5}$ has a horizontal asymptote at $y=3$, the ratio of the leading coefficients 6 and 2.

• Hole (removable discontinuity)

  A single missing point on a graph, at an input where a factor cancels from numerator and denominator.

  Example. $\frac{(x-1)(x+5)}{x-1}$ equals $x+5$ except at $x=1$, so the graph is a line with a hole at $(1,6)$.

• Slant asymptote

  A non-horizontal line that a rational function approaches for large inputs. It appears when the numerator degree is exactly one more than the denominator degree.

  Example. $\frac{x^2+1}{x}=x+\frac{1}{x}$, so the graph approaches the line $y=x$ for large inputs.

• Vertical dilation

  Multiplying every output by a constant $a$, as in $a\cdot f(x)$. It stretches the graph away from the $x$-axis when $|a|>1$ and reflects it when $a < 0$.

  Example. If $f$ has a peak at height 5, then $3f(x)$ has a peak at height 15 at the same input.

• Horizontal translation

  Replacing $x$ with $x-h$, as in $f(x-h)$, which slides the graph $h$ units right when $h>0$ and left when $h < 0$.

  Example. The point $(2,7)$ on $f$ moves to $(5,7)$ on $f(x-3)$, three units to the right.

• Reflection

  A flip of a graph over an axis. $-f(x)$ flips it over the $x$-axis and $f(-x)$ flips it over the $y$-axis.

  Example. The point $(4,1)$ on $f$ becomes $(4,-1)$ on $-f(x)$ and $(-4,1)$ on $f(-x)$.

• Polynomial long division

  A way to rewrite $\frac{p(x)}{d(x)}$ as a quotient plus a remainder over $d(x)$. The quotient shows the end behavior of a rational function.

  Example. $\frac{x^2+3x+5}{x+1}=x+2+\frac{3}{x+1}$, so the graph approaches the line $y=x+2$.

• Binomial Theorem

  A rule for expanding $(a+b)^n$: the terms are $\binom{n}{k}a^{n-k}b^k$, with the coefficients from a row of Pascal's triangle.

  Example. $(x+1)^3=x^3+3x^2+3x+1$, using the row 1, 3, 3, 1 of Pascal's triangle.