AP Precalculus Review (Part 1)

A function f is concave up if…

The rates of change of f are increasing

A function f is concave down if…

The rates of change of f are decreasing

A function f is increasing on an interval if…

As the input values increase, the output values always increase.
OR

For all 𝑎 and 𝑏 in the interval, if 𝑎 < 𝑏, then 𝑓(𝑎) < 𝑓(𝑏).

A function f is decreasing on an interval if…

As the input values increase, the output values always decrease.

OR

For all 𝑎 and 𝑏 in the interval, if 𝑎 < 𝑏, then 𝑓(𝑎) > 𝑓(𝑏).

The slope of a function at any given point gives…

The rate of change of the function at that input

Average rate of change of 𝑓 on the interval [𝑎,𝑏]

(𝑓(𝑏)−𝑓(𝑎))/(𝑏−𝑎)

A positive rate of change indicates that the function output is…

Increasing

A negative rate of change indicates that the function output is…

Decreasing

Point of inflection

Point on the graph of a function where the concavity changes, indicating a maximum or minimum rate of change

One-to-one function

Function where each input has a unique output (no repeated outputs)

A relative minimum occurs when a function 𝑓…

Changes from decreasing to increasing

A relative maximum occurs when a function 𝑓…

Changes from increasing to decreasing

Absolute Maximum

The greatest output of a function

Absolute Minimum

The least output of a function

Multiplicity

The number of times a factor occurs in a polynomial function

A polynomial of degree 𝑛 has…

•Exactly 𝑛 complex zeros (real or imaginary)
•Constant 𝑛th differences

•At most 𝑛−1 extrema

If 𝑥=𝑎 is a real zero of a polynomial with an odd multiplicity, then…

The graph of the polynomial passes through the 𝑥-axis at 𝑥 = 𝑎.

If 𝑥 = 𝑎 is a real zero of a polynomial with an even multiplicity, then…

The graph of the polynomial is tangent to the 𝑥-axis at 𝑥 = 𝑎.

Odd Function

𝑓(−𝑥)=−𝑓(𝑥)

Even Function

𝑓(−𝑥)=𝑓(𝑥)

End behavior of a polynomial 𝑓 with an even degree and a negative leading coefficient

lim𝑥→∞𝑓(𝑥)=−∞

lim𝑥→−∞𝑓(𝑥)=−∞

End behavior of a polynomial 𝑓 with an odd degree and a positive leading coefficient

lim𝑥→∞𝑓(𝑥)=∞

lim𝑥→−∞𝑓(𝑥)=−∞

End behavior of a polynomial 𝑓 with an odd degree and a negative leading coefficient

lim𝑥→∞𝑓(𝑥)=−∞

lim𝑥→−∞𝑓(𝑥)=∞

End behavior of a polynomial 𝑓 with an even degree and a positive leading coefficient

lim𝑥→∞𝑓(𝑥)=−∞

lim𝑥→−∞𝑓(𝑥)=∞

If a rational function, 𝑓, has a horizontal asymptote at 𝑦=𝑏, then…

The ratio of leading terms is a constant, 𝑏, lim𝑥→∞𝑓(𝑥)=𝑏, and lim𝑥→−∞𝑓(𝑥)=𝑏

To determine the end behavior of a rational function…

Analyze the ratio of leading terms

A rational function has a zero at 𝑥=𝑎 if…

𝑥 = 𝑎 is a zero of the numerator but NOT the denominator

A rational function has a hole at 𝑥=𝑎 if…

𝑥 = 𝑎 is a zero of the numerator AND the denominator

A rational function has a vertical asymptote at 𝑥=𝑎 if…

𝑥=𝑎 is a zero of the denominator but NOT the numerator

For rational functions, a slant asymptote occurs when…

The degree of the numerator is exactly one more than the degree of the denominator

If a rational function, 𝑓, has a vertical asymptote at 𝑥=𝑎, then lim𝑥→𝑎−𝑓(𝑥)=_____ and lim𝑥→𝑎+𝑓(𝑥)=_______.

±∞ ; ±∞

If a rational function, 𝑓, has a hole at (𝑎,𝐿) then lim𝑥→𝑎−𝑓(𝑥)=lim𝑥→𝑎+𝑓(𝑥)=____.

L

A function 𝑓(𝑥)=𝑎𝑏^𝑥 demonstrates exponential growth if…

𝑏 > 1

A function 𝑓(𝑥)=𝑎𝑏^𝑥 demonstrates exponential decay if…

0 < 𝑏 < 1

Key features of 𝑦=log𝑏𝑥 where 𝑏>1
(that is log base b of x)

•Domain: 𝑥>0

•Range: all real numbers

•Vertical asymptote at x = 0

•Increasing and concave down over entire domain

Key features of 𝑦=𝑏^𝑥 where 𝑏>1

•Domain: all real numbers

•Range: 𝑦>0

•Horizontal asymptote at 𝑦=0

•Increasing and concave up over entire domain