Precalculus

78 flashcards by MentisLibrary

The bridge to calculus: transformations, polynomials, rational functions, conics, sequences, complex numbers, and vectors. The groundwork calculus expects you to already have.

How do you find the domain of a function algebraically?

Exclude inputs that make a denominator zero, put a negative under an even root, or make a log argument non-positive

How do you tell whether a function is even, odd, or neither?

Even if $f(-x) = f(x)$, odd if $f(-x) = -f(x)$, neither otherwise

What does $y = f(x) + k$ do to a graph?

Shifts it vertically by $k$: up if $k > 0$, down if $k < 0$

What does $y = f(x - h)$ do to a graph?

Shifts it horizontally by $h$: right if $h > 0$, left if $h < 0$

What do $y = -f(x)$ and $y = f(-x)$ do to a graph?

$-f(x)$ reflects across the $x$-axis, $f(-x)$ reflects across the $y$-axis

What does $y = af(x)$ do for $|a| > 1$ versus $0 < |a| < 1$?

$|a| > 1$ stretches vertically, $0 < |a| < 1$ compresses vertically

What does $y = f(bx)$ do to a graph?

Compresses horizontally by a factor of $\dfrac{1}{b}$ when $|b| > 1$, and stretches when $|b| < 1$

In what order should transformations be applied?

Horizontal shift and stretch inside the function first, then vertical stretch, reflection, and shift outside

What is the difference between $|f(x)|$ and $f(|x|)$?

$|f(x)|$ reflects the below-axis parts upward. $f(|x|)$ mirrors the right half onto the left

What is the average rate of change of $f$ on $[a, b]$?

$\dfrac{f(b) - f(a)}{b - a}$, the slope of the secant line

What is the difference quotient?

$\dfrac{f(x+h) - f(x)}{h}$

What is the domain of a composite function $f(g(x))$?

All $x$ in the domain of $g$ whose output $g(x)$ is in the domain of $f$

What is the horizontal line test?

If no horizontal line meets the graph more than once, the function is one to one and has an inverse function

How are the graphs of $f$ and $f^{-1}$ related?

They are reflections of each other across the line $y = x$

How do you determine the end behavior of a polynomial?

Look at the leading term: the degree's parity sets whether the ends match, and the leading coefficient's sign sets the direction

What does the multiplicity of a root tell you about the graph?

Odd multiplicity means the graph crosses the axis, even multiplicity means it touches and turns around

What is the Intermediate Value Theorem?

If $f$ is continuous on $[a,b]$ and $k$ lies between $f(a)$ and $f(b)$, then $f(c) = k$ for some $c$ in $[a,b]$

What is the Fundamental Theorem of Algebra?

Every polynomial of degree $n \ge 1$ has exactly $n$ complex roots, counting multiplicity

What is the complex conjugate root theorem?

If a polynomial with real coefficients has root $a+bi$, then $a-bi$ is also a root

What is Descartes' rule of signs?

The number of positive real zeros equals the number of sign changes in $f(x)$, or less by an even number

Where does a rational function have a vertical asymptote?

Where the denominator is zero and the numerator is not, after cancelling common factors

What are the horizontal asymptote rules for rational functions?

If the numerator degree is less, $y = 0$. If equal, $y$ is the ratio of leading coefficients. If greater, there is none

When does a rational function have a slant asymptote?

When the numerator degree is exactly one more than the denominator's. Divide to find it

What creates a hole in a rational function's graph?

A factor that cancels from both numerator and denominator

How do you solve a polynomial inequality?

Move everything to one side, find the zeros, then test the sign on each interval between them

What extra step does a rational inequality require?

Include the zeros of the denominator as boundary points, but never as solutions

What are the key features of $y = b^x$ for $b > 1$?

Domain all reals, range $y > 0$, horizontal asymptote $y = 0$, passing through $(0,1)$, always increasing

What are the key features of $y = \log_b x$ for $b > 1$?

Domain $x > 0$, range all reals, vertical asymptote $x = 0$, passing through $(1,0)$, always increasing

Why must you check solutions to logarithmic equations?

Because the algebra can produce values that make a log argument non-positive

What is the half-life formula?

$A = A_0\left(\dfrac{1}{2}\right)^{t/h}$, where $h$ is the half-life