Jon Jon's AP Pre-Calculus Flashcards

What does the coefficient a do in g(x)=af(b(x+h))+k

|a|>1: Vertical Stretch.

0<|a|<1: Vertical Compression.

a<0: Reflection across the x-axis.

AP Tip: Outside the function affects y-values.

What does h do in g(x)=af(b(x+h))+k?

x+h shifts left.

x-h shifts right.

Memory Trick: Inside = Opposite.

What does k do in a transformed function?

Positive k shifts up.

Negative k shifts down.

AP Tip: Outside = Same Direction.

What does b do in g(x)=af(b(x+h))+k?

Horizontal scale factor is 1/|b|.

If b<0, reflect across the y-axis.

Common mistake: forgetting the reciprocal.

What is Average Rate of Change?

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

Represents the slope of a secant line and the average change between two points.

When is a function positive or negative?

Positive when the graph is above the x-axis.

Negative when the graph is below the x-axis.

What defines an increasing function?

As x increases, y increases. Graph rises from left to right.

What defines a decreasing function?

As x increases, y decreases. Graph falls from left to right.

What is a Point of Inflection?

A point where concavity changes from up to down or from down to up.

What justifies an increasing rate of change?

Concave Up. Slopes become increasingly positive.

What justifies a decreasing rate of change?

Concave Down. Slopes become less positive or more negative.

What happens at a zero with odd multiplicity?

The graph crosses the x-axis.

Examples: multiplicity 1, 3, 5.

What happens at a zero with even multiplicity?

The graph touches the x-axis and turns around.

Examples: multiplicity 2, 4, 6.

What is an even function?

f(-x)=f(x). Symmetric about the y-axis.

What is an odd function?

f(-x)=-f(x). Symmetric about the origin.

What does lim(x→-∞)f(x) describe?

Left end behavior. Follow the graph left forever.

What does lim(x→∞)f(x) describe?

Right end behavior. Follow the graph right forever.

When does a rational function have y=0 as a horizontal asymptote?

When the degree of the numerator is less than the degree of the denominator.

How do you find a horizontal asymptote when degrees are equal?

Use the ratio of the leading coefficients.

When does a rational function have no horizontal asymptote?

When the degree of the numerator is greater than the degree of the denominator.

When does a rational function have a slant asymptote?

When the numerator degree is exactly one greater than the denominator degree. Use long division.

What causes a hole in a rational function?

A common factor cancels from the numerator and denominator.

What causes a vertical asymptote?

A denominator factor remains after simplifying and equals zero.

What defines an arithmetic sequence?

A constant difference between consecutive terms.

What defines a geometric sequence?

A constant ratio between consecutive terms.

What is exponential decay?

0<b<1. Outputs approach the x-axis as x increases.

What is exponential growth?

b>1. Outputs move away from the x-axis as x increases.

What is the Product Rule for logarithms?

log_b(x)+log_b(y)=log_b(xy).

What is the Power Rule for logarithms?

nlog_b(x)=log_b(x^n).

What is the Quotient Rule for logarithms?

log_b(x)-log_b(y)=log_b(x/y).

What is the Change of Base Formula?

log_a(b)=log(b)/log(a).

What is the Pythagorean Identity?

sin²(x)+cos²(x)=1.

What is the Sine Sum Formula?

sin(A+B)=sin(A)cos(B)+cos(A)sin(B).

What is the Sine Difference Formula?

sin(A-B)=sin(A)cos(B)-cos(A)sin(B).

What is the Cosine Sum Formula?

cos(A+B)=cos(A)cos(B)-sin(A)sin(B).

What is the Cosine Difference Formula?

cos(A-B)=cos(A)cos(B)+sin(A)sin(B).

Where does secant have vertical asymptotes?

Where cos(x)=0.

Where does cosecant have vertical asymptotes?

Where sin(x)=0.

Where does cotangent have vertical asymptotes?

Where sin(x)=0.

Where does tangent have vertical asymptotes?

Where cos(x)=0.

What is the period of sine and cosine functions?

2π/|b|.

What is the period of tangent and cotangent functions?

π/|b|.

How do you convert polar coordinates to rectangular coordinates?

x=r cos(θ), y=r sin(θ).

How do you convert rectangular coordinates to polar coordinates?

r=√(x²+y²), tan(θ)=y/x.

Always check quadrant.

How do you write a+bi in polar form?

r(cos(θ)+i sin(θ)), where r=√(a²+b²).

What is the arc length formula?

s=rθ, where θ is measured in radians.

What is the range of arcsin(x)?

[-π/2, π/2].

What is the range of arccos(x)?

[0, π].

What is the range of arctan(x)?

(-π/2, π/2).

When is distance from the origin increasing in a polar graph?

When r is positive and increasing OR negative and decreasing.

When is distance from the origin decreasing in a polar graph?

When r is positive and decreasing OR negative and increasing.

What is error in a regression model?

|Predicted Value - Actual Value|.

Smaller error indicates a better model.

What is the difference between period and frequency?

Period is the length of one cycle.

Frequency is the number of cycles per unit time.

Frequency = 1/Period.

How can you identify a function from a table?

Constant 1st differences: Linear.

Constant 2nd differences: Quadratic.

Constant 3rd differences: Cubic.

Constant ratio: Exponential.

How can you distinguish exponential and logarithmic functions?

Exponential functions have constant output ratios.

Logarithmic functions have constant input ratios.

They are inverses and reflect across y=x.