Ap pre calc flashcard study guide

A function is concave up when:

Rate of change is increasing

A function is concave down when:

Rate of change is decreasing

A function is increasing on an interval if:

As the input values increase, the output valuesalso increase in that interval.

A function is decreasing on an interval when:

As input values increase, the output values decrease

Average rate of change on an interval formula

f(b)-f(a)/b-a

The slope of a function at any given point gives

The rate of change of the function

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:

A 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

A relative minimum occurs when a function

Changes from decreasing to increasingA

A relative maximum occurs when a function:

Changes from increasing to decreasing

Absolute minimum:

The least output of a function (not negative infinity)

Absolute maximum:

The greatest output of a function (not infinity)

Multiplicity

The number of times a factor occurs in a polynomial function

A polynomial of degree n has:

Exactly n complete zeroes (real or imaginary)

If x=a is a real zero of 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 (bounces off the x-axis)

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

lim f(x) = −∞

x→∞

lim f(x) = −∞

x→-∞

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

lim f(x) = ∞

x→∞

lim f(x) = −∞

x→-∞

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

lim f(x) = −∞

x→∞

lim f(x) = ∞

x→-∞

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

lim f(x) = ∞

x→∞

lim f(x) = ∞

x→-∞

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

The ratio of leading terms

is a constant, b,

lim f(x) = b

x→∞

and,

lim f(x) = b

x→-∞

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 lim f(x) = _______

x→a^-

lim f(x) = _______

x→a⁺

±∞ ; ±∞

If a rational function, f,

has a hole at (a, L) then

lim f(x) = ____.

x→a^-

lim f(x) = ____.

x→a⁺

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 = log_bx where b > 1

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

b^x+c =

b^x ⋅ b^c

b^x-c =

b^x/b^c

e^{a ln b} =

b^a

log_b(1) =

0

log_b(b) =

1

log_b(mn)

log_b m + log_b n

log_bm^k=

k log_bm

log_b(m/n)

log_b m − log_b n

Pythagorean Identities

sin² θ + cos² θ = 1

1 + cot² θ = csc²θ

tan² θ + 1 = sec² θ

sec θ =

csc θ =

cot θ =

1/cos θ

1/sin θ

1/tan θ

=cos θ/sin θ

sin(α ± θ)

sin α cos θ ± sin θ cos α

cos(α ± θ)

cos α cos θ ∓ sin α sin θ

sin(2θ)

2 sin θ cos θ

cos(2θ)

cos² θ − sin² θ

= 2 cos² θ − 1

= 1 − 2 sin² θ

Given (x, y) in Cartesian (rectangular) coordinates, determine polar coordinates, (r, θ)

r = root x² + y²

θ = tan^-1 (y/x)

(*Add π if angle is in Q2 or Q3)

Given (r, θ) in polar coordinates, determine Cartesian coordinates, (x, y)

x = r cos θ

y = r sin θ

A polar function r = f(θ) is decreasing if...

As θ increases, r decreases.

A polar function r = f(θ) is increasing if...

As θ increases, r increases.

The distance between a point on a polar function r = f(θ) and the origin is decreasing if...

r is positive and decreasing or r is negative and increasing. (|r| is decreasing)

The distance between a point on a polar function r = f(θ) and the origin is increasing if...

r is positive and increasing or r is negative and decreasing. (|r| is increasing)

A function is linear if over equal-length input intervals, output values ______.

Change by a constant amount.

A function is quadratic if over equal-length input intervals, output values ______.

Change by a constant second difference.

A function is exponential if as input values change ____, output values change ____.

additively; multiplicatively

A function is logarithmic

if as input values change ____, output values change _____.

multiplicatively; additively

The average rates of change of a linear function are...

Constant

The average rates of change of a quadratic function...

Are changing at a constant rate OR follow a linear pattern

tan θ gives the ______ of the terminal ray of θ.

Slope

Domain and range of y = arcsin x

Domain: [−1,1]

Range: [-π/2,π/2)

Domain and range of y = arccos x

Domain: [−1,1]

Range: [0, π]

Domain and range of y = arctan x

Domain: (−∞, ∞)

Range: (-π/2,π/2)

f(x) = tan x has vertical asymptotes at...

x =

π/2 + πk, where k is an integer

f(x) = cot x has vertical asymptotes at ...

x = πk, where k is an integer

Determine the amplitude, period,

midline, and phase shift of

f(x) = a sin(b(x − c)) + d

Amplitude = |a|

Period = 2π/b

Midline: y = d

Phase shift: c units to the right

y = tan(bx) has a period of...

π/b

Key features of y = sin x

• Domain: all real numbers

• Range: [–1, 1]

• Period: 2π

• Amplitude: 1

• Midline: y = 0

• Passes through (0, 0)

Key features of y = cos x

• Domain: all real numbers

• Range: [–1, 1]

• Period: 2π

• Amplitude: 1

• Midline: y = 0

• Passes through (0, 1)

f(x) + c

Vertical translation c units up if c > 0 or c units down if c < 0

f(x − c)

Horizontal translation c units to the right if c > 0 or c units to the left if c < 0

f(cx)

Horizontal dilation by a

factor of 1/c

cf(x)

Vertical dilation by a factor of c

−f(x)

Reflection over the x-axis

f(−x)

Reflection over the y-axis

What does the constant e represent?

The base rate of growth for all continually growing processes e ≈ 2.718

Pascal’s Triangle

A positive residual indicates that the predicted value is an ___________.

Underestimate

A negative residual indicates that the predicted value is an ___________.

Overestimate

Explicit rule for nth term of a geometric sequence given common ratio r, and the a_k term

a_n = a_k ⋅ r^n-k

Explicit rule for nth term of an arithmetic sequence given common difference d, and the a_k term

a_n = a_k + d(n − k)

A model is considered appropriate for a data set if the residual plot...

Appears without pattern

Residual

Actual value – Predicted value

Error (in a model)

Predicted value – Actual value

f and g are inverse functions if...

f(g(x)) = g(f(x)) = x

In a semi-log plot where the y-axis is logarithmically scaled, exponential functions will appear

linear

If the y-axis is logarithmically scaled, then...

Equal-sized increments on the y-axis represent proportional changes in the output variable

What graph is this?

y=csc x

What graph is this?

y=sec x

What graph is this?

y=cot x

What graph is this?

y=arcsin x

What graph is this?

y=arccos x

What graph is this?

y=arctan x

How does this graph occur?

When b>a

How does this graph occur?

When a>b

How does this graph occur?

When a=b