AP Precalculus

One-to-one function

each x value pairs to exactly one UNIQUE y value

- Passes HLT & VLT

(No repeated x or y values in a table)

Function

each x value pairs to exactly one y value

- Passes VLT

(No repeated x values in a table)

Degree of a polynomial

the greatest degree of any term in the polynomial

Leading term

the term with the highest degree

What is the maximum number of turning points a polynomial can have?

n-1

n = the degree of the polynomial

Inflection point

= Leading exponent - 2

Point where concavity changes

(CU to CD or CD or CU)

Asymptote

a line that a graph approaches but never crosses

Asymptote(s) of an exponential function

horizontal

Asymptote(s) of a logarithmic function

vertical

Vertical stretch

the whole function, f(x), is multiplied by a (where a > 1)

Vertical compression

the whole function, f(x) is multiplied by a (where a is a fraction: 0 < a < 1)

Horizontal stretch

- x is multiplied by a (where a is a fraction: 0 < a < 1)

- graph stretches away from the y-axis

Horizontal compression

- x is multiplied by a (where a > 1)

- graph compresses toward the y-axis

How do you solve an algebraic equation with an absolute value?

1. Isolate the absolute value

2. Solve for a + other side (solution 1)

3. Solve for a - other side (solution 2)

how to find the inverse of a function

1. Change f(x) to y

2. Swap x & y

3. Isolate y

4. y becomes f^-1(x)

A function is even if

f(-x) = f(x)

- It is symmetric over the y-axis

A function is symmetric over the y-axis if

f(-x) = f(x)

- It's an even function

A function is odd if

f(-x) = -f(x)

- It is symmetric over the origin

A function is symmetric over the origin if

f(-x) = -f(x)

- It is an odd function

f^-1(f(x))

x

f(n) = m, f^-1(m) = ___

n

leading coefficient

The coefficient of the term with the highest degree (leading term)

Solve an inequality with an absolute value

1. Isolate the absolute value

2. Solve for a + other side

3. Solve for a - other side AND flip the inequality

Complex zeroes

When solving for all the zeroes, there will be a negative under the square root. Replace it with i

- Its conjugate is ALSO a zero

Number of zeros of a polynomial

= the exponent of the leading term (the highest exponent)

- some can be imaginary and unseen on the graph

Complex conjugate

Complex numbers: a+bi and a-bi

- If a function has a zero at a+bi, it ALSO has a zero at a-bi

Multiplicity of a zero

Number of times a zero's factor occurs in a polynomial

- If odd, the line passes through that zero

- If even, the line will be tangent to the x-axis (bounce off) at that zero

End behavior (EB)

The behavior as x approaches positive or negative infinity:

- EB of an even function is the same for its -∞ & ∞

- EB of an odd function is the opposite for its -∞ & ∞

How to find the equation of a quadratic from a table of values?

- AOS = avg of x values from equal y values

- OR plug two/three points into ax^2 + bx + c form & solve system of equations (a is 2nd difference/2)

Linear, quadratic, or exponential from table of values?

- Linear = First difference in y is constant

- Quadratic = Difference in y is not constant, but the second difference (difference between successive first differences) is constant

* a = second difference / 2

- Exponential = Difference in y follows a similar pattern to the y values

How to find the degree of a function (from graph)

= Number of inflection points + 2

Intermediate Value Theorem

if f(x) is continuous on [a,b], then f(x) passes through every value between f(a) and f(b)

The Polynomial Remainder Theorem

1. With any polynomial -- p(x) -- if p(a) = 0, then x-a is a factor of p(x)

2. If p(a) = y, then y is the remainder when p(x) is divided by x-a

Factor Theorem of Polynomials

A polynomial f(x) has a factor (x-a) if and only if f(a)=0

Rational Zero Theorem

If f(x) = a⌄nx^n + ... +ax + a (a polynomial function) has integer coefficients, then every rational zero of f(x) takes the form:

p/q = factor of a (constant) / factor of a⌄n (leading coefficient)

Rational function

R(x) = f(x)/g(x)

f(x) is a polynomial or monomial/constant

g(x) is a polynomial & not 0

End behavior of a rational function

1. Numerator degree > denominator degree:

- EB matches the EB of the quotient of the leading terms

2. Numerator degree = denominator degree:

- EB approaches the horizontal asymptote (= ratio of leading terms) in both directions

3. Numerator degree < denominator degree:

- EB approaches the horizontal asymptote y = 0 in both directions

Types of Discontinuity

1. Infinite/asymptotic

2. Removable/point

3. Jump

AP Exam: Round to ___ decimal places

3

THE THIRD ONE!

AP exam, you should round to _____ decimal places

3

e =

roughly 2.718

A horizontal translation of an exponential function is...

the same as giving it a vertical stretch or changing its vertical stretch

E.g., if f(x) = b^x, then

f(x+1) = b^(x+1) = b^x * b (in this case, b becomes a, the vertical stretch)

A horizontal stretch/compression of an exponential function is...

the same as changing its base

E.g., f(x) = b^x, then

f(3x) = b^3x = (b^3)^x (effectively changing the base from b to b^3)

A reflection over the y axis of an exponential function is...

the same as making the base its inverse

E.g., if f(x) = b^x, then

f(-x) = b^-x = (1/b)^x

Continuous growth/decay formula

f(t) = a * e^Kt

1. a is the initial amount

2. e is the constant natural base (the letter e, like pi)

3. k the rate of change per unit of time (DON'T add 1 if, e.g., 5%)

4. t is the units of time

Log and exponential functions are _____ of each other

inverse functions

y = log(base b)x is the inverse of...

y = b^x

if log(base b)C = a, then

b^a = C

if b^a = C, then

log(base b)C = a

log (x*y) =

log (x) + log (y)

log (x/y) =

log x - log y

log (x^P) =

P * log (x)

log (3^x) =

x * log (3)

log without a base on a calculator is

log (base 10)

To take the natural log on a calculator?

ln

log(base e) =

ln

if log(base a)x = log(base a)y, then

x = y

if ln(x) = ln(y), then

x = y

log(base a)1 =

0, because a^0 = 1

log(base a)a =

1, because a^1 = a

log(base a)a^x =

x, because a^x = a^x

log(base x) of ≤ 0

does not exist (gives a calculation error)

Natural vs. common log

Natural = base e

Common = base 10

The change of base formula

log(base b)x = log(x)/log(b)

The same is true for any base on the right side as well, including e (the natural log (ln))

How to find the domain/range of the inverse of a function

The range of the inverse function is the domain of the original

The domain of the inverse is the range of the original

What is one (slow but notable) way to find a log function (or inverse function

Create a table of values for the original function (the exponential version), then swap the x and y values to get the inverse (log) function's table of values

Exponential functions are the inverse of

logarithmic functions

When you get solutions to log functions, you need to...

plug them in because they may be extraneous

The 6 things to know about log graphs

1. Domain is positive real #s

2. x-int is (1, 0)

3. y-int DOESN'T EXIST, as log(0) doesn't exist (x = 0 is an asymptote)

4. If base > 1, it's an increasing function

5. If 0 < base < 1, it's a decreasing function

6. Always either increasing or decreasing, meaning no extrema or relative extrema and no points of inflection

Extrema and relative extrema

Extrema: Absolute maxes/mins

Relative extrema: Local maxes/mins

When do you need to plug in your solutions to check for extraneous solutions?

When solving:

1. Radical equations (when you get a solution after squaring both sides to get rid of a square root)

2. Logarithmic equations (with logs)

How to solve a logarithmic equation with a constant (steps)

1. Bring logs with the same base to the same side, and combine into 1 log

2. Get into the form: log(base b)c = a

3. Rewrite in exponential form: b^a = c and solve

4. Check the solution(s) in the original equation (extraneous if it creates a log of zero or a negative number)

If you're modeling a function with continuous exponential or logarithmic change...

The base is e. (For logs, this means use ln).

e^ln(x)=

x for x>0

Continuous exponential change formula

f(x) = A(e)^rt

A = initial amount

r = rate of change per unit of time (e.g., if +1%, it's 1.01)

t = time

When you have to make an exponential function for something in nature (e.g., bacterial growth)...

USE BASE e

e

2.718

Residual

observed value - predicted value

Residual plot

x-axis = independent variable

y-axis = residual values (difference between observed and predicted values)

- If points are randomly dispersed, the model is appropriate

Arithmetic sequence

The difference between consecutive terms is constant (called the common difference)

Term of arithmetic sequence formula

An = Ak + d(n - k)

An = the nth term (you are trying to find)

Ak = the kth term in the sequence

n = the desired term number

k = the given term number (often 1)

d = the common difference

Geometric sequence

The ratio of consecutive terms is constant (called the common ratio)

Term of Geometric sequence formula

An = Ak * r^(n-k)

An = the nth term (you are trying to find)

Ak = the kth term in the sequence

n = the desired term number

k = the given term number (often 1)

r = the common ratio

Finite series

series with a first term and a last term (finite length)

Infinite series

series with a first term but no last term (infinite length)

Sequences vs. functions

sequences are discrete; functions are continuous

Period (english definition)

The domain (x-value interval) required to complete a full cycle of a periodic function

Period (mathematic condition)

A period must be the smallest value K such that f(x+K) = f(x)

- The interval it takes to return to the same value

- Necessary but not sufficient to be a period

Cycle

A complete pattern of y values in a periodic function

Periodic function

A function that repeats a pattern of y-values (cycle) at regular intervals (periods)

tan(45) =

= x/x = 1

Because it means the triangle is right isosceles, so the legs are equal, yielding x/x

Trig functions are ______ functions

periodic

Standard position

An angle with its vertex at the origin and its initial side fixed on the positive x-axis

Initial vs. terminal side

Initial: The fixed side that does not rotate

Terminal: The non-fixed side that rotates

Terminal side rotation

- If it rotates clockwise (starting into negative y values), the angle is negative

- If it rotates counterclockwise (starting into positive y values) the angle is positive

Vertex of an angle

The common endpoint of the two lines that form an angle

A 1 radian center angle in a circle means...

the 1 radian angle's arc length equals the radius

Coterminal angles

Angles in standard position that share a terminal side

- Have the same trig values

- E.g., 50° is coterminal to 360°+50,° meaning

sin/cos/tan(50°) = sin/cos/tan(360°+50°)

- E.g., pi radians is coterminal to pi+2pi radians, meaning

sin/cos/tan(pi) = sin/cos/tan(pi+2pi)

Quadrantal angle ("quadrantals")

An angle in standard position with its terminal side on the x or y axis (e.g., 0, 90, 180, 270, 360)

In the xy-plane, sin θ = ___

sin θ = vertical displacement/r,

where r is the radius