AP Precalculus Notes

Describing Functions

• Positive: y-values are positive; the graph lies above the x-axis. Example: $f(x) = x^2 + 1$ for all real numbers.

• Negative: y-values are negative; the graph lies below the x-axis. Example: $f(x) = -x^2 - 1$ for all real numbers.

• Increasing: Graph goes up as x increases; the function's slope is positive. Example: $f(x) = x$ is increasing everywhere.

• Decreasing: Graph goes down as x increases; the function's slope is negative. Example: $f(x) = -x$ is decreasing everywhere.

• Increasing at an increasing rate: Increase is getting faster; the graph is concave up. Example: $f(x) = x^2$ for x > 0.

• Increasing at a decreasing rate: Increase is getting slower; the graph is concave down. Example: $f(x) = \sqrt{x}$ for x > 0.

• Decreasing at a decreasing rate: Decrease is getting slower; the graph is concave up. Example: $f(x) = -\sqrt{x}$ for x > 0.

• Decreasing at an increasing rate: Decrease is getting faster; the graph is concave down. Example: $f(x) = -x^2$ for x > 0.

Describing Rate of Change

• Rate of change is increasing if concave up; the second derivative is positive. Example: $f(x) = x^2$, f''(x) = 2 > 0.

• Rate of change is decreasing if concave down; the second derivative is negative. Example: $f(x) = -x^2$, f''(x) = -2 < 0.

• Rate of change is positive if f is increasing; first derivative is positive. Example: $f(x) = x$, f'(x) = 1 > 0.

• Rate of change is negative if f is decreasing; first derivative is negative. Example: $f(x) = -x$, f'(x) = -1 < 0.

• Rate of change is zero if f is constant; the tangent line is horizontal. Example: $f(x) = 5$, $f'(x) = 0$.

Average Rate of Change

The average rate of change is calculated as:

$\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$

This represents how much the y-variable changes, on average, per unit of the x-variable. It's the slope of the secant line between two points on the curve. Example: For $f(x) = x^2$ between $x = 1$ and $x = 3$, the average rate of change is $\frac{3^2 - 1^2}{3 - 1} = \frac{8}{2} = 4$.

Instantaneous Rate of Change

• The instantaneous rate of change is the slope of the tangent line at a specific point.

• It can be estimated by finding the average rate of change over a very small interval around that point. This is the limit as $\Delta x$ approaches zero. Example: For $f(x) = x^2$ at $x = 2$, the instantaneous rate of change is $2x = 4$.

Function Types and Differences

• Linear Function: First differences are equal; the function can be written in the form $f(x) = mx + b$. Example: $f(x) = 2x + 3$.

• Quadratic Function: Second differences are equal; the function can be written in the form $f(x) = ax^2 + bx + c$. Example: $f(x) = x^2 + 2x + 1$.

• Cubic Function: Third differences are equal; the function can be written in the form $f(x) = ax^3 + bx^2 + cx + d$. Example: $f(x) = x^3 + 3x^2 + 3x + 1$.

Polynomial Functions

• Degree: The highest power of the variable in the polynomial; determines the end behavior. Example: In $f(x) = x^3 + 2x^2 - x + 5$, the degree is 3.

• Relative Min/Max: Minimum or maximum value compared to the points immediately around it; also known as local extrema. Example: $f(x) = x^3 - 3x$ has a relative max at $x = -1$ and a relative min at $x = 1$.

• Absolute Min/Max: Minimum or maximum value of the entire function; also known as global extrema. Example: $f(x) = x^2$ has an absolute min at $x = 0$.

• Multiplicity: The number of times a factor occurs in the polynomial.

  • Multiplicity 1: Linear factor; the graph crosses the x-axis. Example: $f(x) = (x - 1)$ crosses the x-axis at $x = 1$.

  • Multiplicity 2: Quadratic factor (touches the x-axis and turns around); the graph is tangent to the x-axis. Example: $f(x) = (x - 2)^2$ touches the x-axis at $x = 2$.

  • Multiplicity 3: Cubic factor (has a similar shape to x^3 at the root); the graph has an inflection point at the root. Example: $f(x) = (x - 3)^3$ has an inflection point at $x = 3$.

• Imaginary Zeros: Come in conjugate pairs (e.g., $1 + 2i$ and $1 - 2i$); if a polynomial has real coefficients. Example: If $3 + i$ is a zero, then $3 - i$ is also a zero.

• Number of Zeros: The degree of a polynomial equals the total number of zeros, including multiplicity and imaginary roots; by the Fundamental Theorem of Algebra. Example: A polynomial of degree 4 has 4 zeros.

• Polynomial Form: A polynomial can be written as $a(x - first\ zero)(x - second\ zero)(x - third\ zero)…$ . Example: If zeros are 1, -2, and 3, then $f(x) = a(x - 1)(x + 2)(x - 3)$.

Function Symmetry

• Even Function: Symmetric with respect to the y-axis; $f(x) = f(-x)$. Example: $f(x) = x^2$.

• Odd Function: Symmetric with respect to the origin (180° rotational symmetry); $f(-x) = -f(x)$. Example: $f(x) = x^3$.

End Behavior of Polynomial Functions

The end behavior depends on the leading coefficient and the degree of the polynomial.

• Positive Leading Coefficient:

  • Even Degree: $\lim<em>{x \to -\infty} f(x) = \infty$ and $\lim</em>{x \to \infty} f(x) = \infty$. Example: $f(x) = x^2$.

  • Odd Degree: $\lim<em>{x \to -\infty} f(x) = -\infty$ and $\lim</em>{x \to \infty} f(x) = \infty$. Example: $f(x) = x^3$.

• Negative Leading Coefficient:

  • Even Degree: $\lim<em>{x \to -\infty} f(x) = -\infty$ and $\lim</em>{x \to \infty} f(x) = -\infty$. Example: $f(x) = -x^2$.

  • Odd Degree: $\lim<em>{x \to -\infty} f(x) = \infty$ and $\lim</em>{x \to \infty} f(x) = -\infty$. Example: $f(x) = -x^3$.

Rational Functions

A rational function $h(x)$ is defined as $\frac{f(x)}{g(x)}$ where both $f(x)$ and $g(x)$ are polynomial functions.

• Degree on Top Bigger: Unbounded, no horizontal asymptote. Simplify the ratio of leading terms to determine end behavior. Example: $\frac{x^2}{x}$.

  • Slant Asymptote: If the degree on top is exactly one more than the degree on the bottom, find the slant asymptote by dividing $f(x)$ by $g(x)$. The quotient (without the remainder) gives the equation of the slant asymptote. Example: For $\frac{x^2 + 1}{x}$, the slant asymptote is $y = x$.

• Degree on Bottom Bigger: Horizontal asymptote at $y = 0$. Example: $\frac{x}{x^2}$.

• Same Degrees: Horizontal asymptote at y = \frac{ratio \ of \ leading \ coefficients}. Example: $\frac{2x^2}{3x^2}$ has a horizontal asymptote at $y = \frac{2}{3}$.

• To analyze rational functions further:

  • Vertical Asymptote: Occurs where the factor is only on the bottom; set the denominator equal to zero and solve. Example: In $\frac{1}{x - 2}$, the vertical asymptote is at $x = 2$.

  • X-intercepts: Occur where the factor is only on the top; set the numerator equal to zero and solve. Example: In $\frac{x - 3}{x + 1}$, the x-intercept is at $x = 3$.

  • Hole: Occurs when a factor is present on both the top and bottom; simplify the function after removing the common factor. Example: $\frac{(x - 1)(x + 2)}{(x - 1)}$ has a hole at $x = 1$.

  • Y-intercept: Found by substituting $x = 0$ into the function. Example: In $\frac{x + 1}{x + 2}$, the y-intercept is $\frac{1}{2}$.

Binomial Expansion

$(a + b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1}b^1 + \binom{n}{2}a^{n-2}b^2 + … + \binom{n}{n}a^0 b^n$

Coefficients can be found using Pascal's Triangle.

• The power of $a$ decreases from $n$ to 0.

• The power of $b$ increases from 0 to $n$.

Example: $(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$

Function Transformations

• Vertical Translation:

  • $f(x) + c$: Up when c > 0, down when c < 0. Example: $f(x) = x^2 + 3$ (up 3 units), $f(x) = x^2 - 3$ (down 3 units)

• Horizontal Translation:

  • $f(x + c)$: Right when c < 0, left when c > 0. Example: $f(x) = (x + 3)^2$ (left 3 units), $f(x) = (x - 3)^2$ (right 3 units)

• Vertical Dilation:

  • $c \cdot f(x)$

  • Dilation factor: $|c|$

  • Stretch when |c| > 1. Example: $f(x) = 2x^2$ (vertical stretch)

  • Shrink when 0 < |c| < 1. Example: $f(x) = 0.5x^2$ (vertical shrink)

• Horizontal Dilation:

  • $f(cx)$

  • Dilation factor: $\frac{1}{|c|}$

  • Stretch when 0 < |c| < 1. Example: $f(x) = (0.5x)^2$ (horizontal stretch)

  • Shrink when |c| > 1. Example: $f(x) = (2x)^2$ (horizontal shrink)

• Reflection over x-axis:

  • $-f(x)$: Reflect vertically. Example: $f(x) = -x^2$

• Reflection over y-axis:

  • $f(-x)$: Reflect horizontally. Example: $f(x) = (-x)^3 = -x^3$

Solving for Parameters Algebraically

Plug in given $(x, y)$ pairs and solve the system of equations using substitution, elimination, or graphing. Ensure you have as many equations as unknowns. Example: Given $f(x) = ax + b$, and points (1, 5) and (2, 7), solve for a and b.

Arithmetic Sequences

• Repeated addition.

• Linear function where the domain is positive integers.

• Formulas:

  • $y = mx + b = b + mx$

  • $a<em>n = a</em>1 + d(n - 1)$

  • $a<em>n = a</em>2 + d(n - 2)$

• $d$ is the common difference. Example: 2, 4, 6, 8, … (d = 2)

Geometric Sequences

• Repeated multiplication.

• Exponential function where the domain is positive integers.

• Formulas:

  • $y = ab^x$

  • $a<em>n = a</em>0 r^n$

  • $a<em>n = a</em>1 r^{n-1}$

  • $a<em>n = a</em>2 r^{n-2}$

• $r$ is the common ratio. Example: 2, 4, 8, 16, … (r = 2)

Exponential Functions

• Form: $f(x) = ab^x$

  • $a$ is the initial value.

  • $b$ is the growth factor.

  • If 0 < b < 1, exponential decay.

  • If b > 1, exponential growth.

• If change is given as a percent, convert to a decimal and add (for growth) or subtract (for decay) from 1 to get $b$. For example, a 5% increase means $b = 1.05$, and a 5% decrease means $b = 0.95$. Example: $f(x) = 2(1.05)^x$ (5% growth), $f(x) = 2(0.95)^x$ (5% decay)

Compound Interest

• $P(t) = P_0 (1 + \frac{r}{n})^{nt}$

  • $P_0$: Initial amount.

  • $r$: Interest rate (as a decimal).

  • $n$: Number of compoundings per year (monthly, weekly, yearly, etc.).

  • $t$: Number of years.

Example: $\$1000$ invested at 5% compounded monthly for 10 years: $P(10) = 1000(1 + \frac{0.05}{12})^{12 \cdot 10}$

Compounding Continuously

• $P(t) = P_0 e^{rt}$

Example: $\$1000$ invested at 5% compounded continuously for 10 years: $P(10) = 1000e^{0.05 \cdot 10}$

Half-Life Problems

• $f(t) = Initial \cdot (\frac{1}{2})^{\frac{t}{half-life}}$

Example: 50 grams of a substance with a half-life of 10 years: $f(t) = 50(\frac{1}{2})^{\frac{t}{10}}$

Regression

• Using a calculator:

  • Stat -> Edit -> Enter data.

  • Stat -> Calc -> Choose regression.

  • Calculate.

• Residual: Actual - Predicted.

• Good models have no pattern in the residual plot; the residuals should be randomly scattered. Example: If actual value is 5 and predicted is 4.8, the residual is 0.2

Composition of Functions

• $g(f(x)) = (g \circ f)(x)$

• Evaluate from inside out; $f(x)$ is inside $g(x)$. The output of $f(x)$ becomes the input of $g(x)$. Example: If $f(x) = x + 1$ and $g(x) = x^2$, then $g(f(x)) = (x + 1)^2$

Inverse Functions

• Inverse functions undo original functions. To find them, solve for the other variable (swap input and output). Check by verifying $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. Example: If $f(x) = 2x$, then $f^{-1}(x) = \frac{x}{2}$

Invertibility

• A function $f$ is invertible if it has an inverse function.

• To check if $f$ is invertible, make sure $f$ is one-to-one (passes the horizontal line test). One-to-one means each $y$ value corresponds to exactly one $x$ value. Example: $f(x) = x^2$ is not invertible over all real numbers but it's invertible when its domain is restricted to only positive real numbers.

Exponential and Logarithmic Functions

• Exponentials and logs are inverses.

• $\\log_b x = y \iff b^y = x$

Example: $\log_2 8 = 3 \iff 2^3 = 8$

Exponential Function

• Form: $f(x) = b^x$

• Domain: All real numbers ($\mathbb{R}$).

• Range: y > 0

• Horizontal Asymptote: $y = 0$

• Passes through the points $(0, 1)$ and $(1, b)$. This function represents exponential growth if b > 1 and exponential decay if 0 < b < 1. Example: $f(x) = 2^x$

Logarithmic Function

• Domain: x > 0

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