AP Pre-Calc Unit 1 Summary Flashcards

What is a polynomial function?

A function made of terms with real coefficients and nonnegative integer exponents.

What is the degree of a polynomial?

The highest exponent of x after simplifying.

What is the leading coefficient?

The coefficient of the highest-degree term.

What is the leading term?

The term containing the highest power of x.

What is a constant polynomial?

A polynomial containing only a constant; a nonzero constant has degree 0.

What is a zero of a function?

An x-value that makes f(x)=0.

What is an x-intercept?

A point where a graph intersects the x-axis, so y=0.

How do you find polynomial zeros?

Set f(x)=0 and solve, often by factoring.

What is multiplicity?

The number of times a zero appears as a factor.

What happens at an odd-multiplicity zero?

The graph crosses the x-axis.

What happens at an even-multiplicity zero?

The graph touches the x-axis and turns around.

What happens at multiplicity 1?

The graph crosses the x-axis without flattening.

What happens at multiplicity 2?

The graph touches the x-axis and bounces.

What happens at multiplicity 3?

The graph crosses the x-axis while flattening.

How does higher multiplicity affect a graph?

The graph becomes flatter near that zero.

How do you find zeros from factored form?

Set each factor equal to zero.

What does (x−4)² indicate?

A zero at x=4 with multiplicity 2.

What is polynomial end behavior?

How the graph behaves as x approaches positive or negative infinity.

What determines polynomial end behavior?

The degree and leading coefficient.

Even degree with a positive leading coefficient?

Both ends rise.

Even degree with a negative leading coefficient?

Both ends fall.

Odd degree with a positive leading coefficient?

Left end falls, right end rises.

Odd degree with a negative leading coefficient?

Left end rises, right end falls.

What is the maximum number of turning points for degree n?

n−1 turning points.

How many complex zeros does a degree-n polynomial have?

Exactly n, counting multiplicity.

What is the Fundamental Theorem of Algebra?

Every nonconstant polynomial has at least one complex zero.

What is a complex number?

A number in the form a+bi, where a and b are real.

What is the imaginary unit i?

i²=−1.

How can a real number be written in complex form?

a+0i.

What is a nonreal complex number?

a+bi, where b≠0.

What is a complex conjugate?

A number formed by reversing the sign of the imaginary part.

What is the conjugate of −3+6i?

−3−6i.

What is the conjugate zeros theorem?

Nonreal zeros of real-coefficient polynomials occur in conjugate pairs.

If 4−2i is a zero, what other zero must exist?

4+2i, assuming real coefficients.

What is an even function?

A function satisfying f(−x)=f(x).

What symmetry do even functions have?

Symmetry across the y-axis.

What is an odd function?

A function satisfying f(−x)=−f(x).

What symmetry do odd functions have?

Symmetry about the origin.

How do you test whether a function is even?

Substitute −x and check whether f(−x)=f(x).

How do you test whether a function is odd?

Substitute −x and check whether f(−x)=−f(x).

Is x⁶−4x²−7 even or odd?

Even, because f(−x)=f(x).

Is −2x³+5x even or odd?

Odd, because f(−x)=−f(x).

Are all even-degree polynomials even functions?

No. Every nonconstant term must have an even exponent.

Can a function be neither even nor odd?

Yes, if neither symmetry condition holds.

What is a rational function?

A quotient of polynomials, f(x)=p(x)/q(x), where q(x)≠0.

How do you find the domain of a rational function?

Exclude x-values that make the original denominator zero.

What is a vertical asymptote?

A vertical line approached by a graph as its values become unbounded.

How do you find vertical asymptotes?

Set the uncanceled denominator factors equal to zero.

What causes a hole in a rational graph?

A factor that cancels from the numerator and denominator.

How do you find a hole's x-coordinate?

Set the canceled factor equal to zero.

How do you find a hole's y-coordinate?

Substitute its x-coordinate into the simplified function.

What is a horizontal asymptote?

A horizontal line the graph approaches as x approaches positive or negative infinity.

What if the numerator degree is smaller than the denominator degree?

The horizontal asymptote is y=0.

What if the numerator and denominator have equal degrees?

The horizontal asymptote is the ratio of their leading coefficients.

What if the numerator degree exceeds the denominator degree?

There is no horizontal asymptote.

When does a rational function have a slant asymptote?

When the numerator degree is exactly one greater than the denominator degree.

How do you find a slant asymptote?

Divide the numerator by the denominator and use the linear quotient.

Can a graph cross a horizontal asymptote?

Yes. Horizontal asymptotes describe end behavior.

Can a graph cross a vertical asymptote?

No. The function is undefined at that x-value.

How do you find rational-function x-intercepts?

Set the simplified numerator equal to zero.

How do you find a y-intercept?

Evaluate f(0), if defined.

What is a parent function?

The basic function before transformations.

How does f(x)+k transform a graph?

Shifts it vertically by k units.

How does f(x−h) transform a graph?

Shifts it h units right when h>0.

How does f(x+h) transform a graph?

Shifts it h units left when h>0.

How does −f(x) transform a graph?

Reflects it across the x-axis.

How does f(−x) transform a graph?

Reflects it across the y-axis.

What does af(x) do when a>1?

Vertically stretches the graph by factor a.

What does af(x) do when 0<a<1?

Vertically compresses the graph by factor a.

What does f(bx) do when b>1?

Horizontally compresses the graph by factor 1/b.

What does f(bx) do when 0<b<1?

Horizontally stretches the graph by factor 1/b.

What is the domain of a function?

The set of all allowable input values.

What is the range of a function?

The set of all possible output values.

What is interval notation?

A way to represent sets of numbers using intervals.

What do parentheses mean in interval notation?

The endpoint is excluded.

What do square brackets mean in interval notation?

The endpoint is included.

How do you express all real numbers in interval notation?

(−∞,∞).

What is an increasing interval?

An interval where f(x) increases as x increases.

What is a decreasing interval?

An interval where f(x) decreases as x increases.

What is a constant interval?

An interval where f(x) stays the same.

What is the average rate of change formula?

[f(b)−f(a)]/(b−a).

What does average rate of change measure?

The slope of the secant line between two points.

What is a secant line?

A line passing through two points on a graph.

What does a positive average rate of change indicate?

The output increased over the interval.

What does a negative average rate of change indicate?

The output decreased over the interval.

What is a removable discontinuity?

A hole that can be eliminated by defining one function value.

What is an infinite discontinuity?

A discontinuity where function values become unbounded.

What is the remainder theorem?

Dividing f(x) by x−c gives a remainder of f(c).

What is the factor theorem?

x−c is a factor of f(x) exactly when f(c)=0.

What is synthetic division?

A shortcut for polynomial division by a linear factor x−c.

What is a polynomial remainder?

The expression left over after polynomial division.

What is a function model?

An equation representing a real-world situation.

What is a linear model?

A model with a constant rate of change.

What is a quadratic model?

A degree-2 model whose graph is a parabola.

What is a cubic model?

A polynomial model of degree 3.

What is the correlation coefficient r?

A measure of the strength and direction of a linear relationship.

What is the possible range of r?

−1≤r≤1.

What does r near 1 indicate?

A strong positive linear association.

What does r near −1 indicate?

A strong negative linear association.

What does r near 0 indicate?

Little or no linear association.