AP Precalculus Review Notes

Topic 1.1: Change in Tandem

• Deals with input and output values.
• Function: A mathematical relation where each input (x) maps to exactly one output (y).
  • Input values: Domain of a function, independent variable.
  • Output values: Range of a function, dependent variable.
• Increasing Function: As input values increase, output values increase.
• Decreasing Function: As input values increase, output values decrease.
• Graph of a Function: Displays input-output pairs, showing how input and output values vary.
• Rate of Change: A graph's slope.
• Concave Up: Rate of change is increasing, looks like a "u".
• Concave Down: Rate of change is decreasing, looks like an upside-down "u".
• Zeros of a Function: Where the graph intersects the x-axis (output value or y value is zero).
• Equation: $y = mx + b$
  • $x$: Input
  • $y$: Output
  • $m$: Slope, rate of change
  • $b$: y-intercept (where graph crosses the y-axis)

Topic 1.2: Rate of Changes

• Rate of change is a graph's slope.
• Slope equation: $\frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
• Finding Average Rate of Change:
  • Given two points on a graph, label them as $(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$.
  • Plug the values into the slope equation.
  • Solve to get the average rate of change between those points.
• Predicting Points Using Average Rate of Change:
  • Multiply the given value by the average rate of change.
• Positive Rate of Change: As one quantity increases or decreases, the other does the same.
• Negative Rate of Change: As one quantity increases, the other decreases.

Topic 1.3: More on Rate of Change

• Linear Function: The rate of change is constant over any interval; average rate of change is unchanging (rate of zero).
• Quadratic Functions: The average rate of change varies linearly.
  • Calculating the average rate of change over an interval calculates the slope of the secant line within that interval.
  • The average rate of change will never be accurate to the actual quadratic function.
• Calculating Average Rate of Change at a Specific Point (e.g., $x = 5$):
  • Find the $y$ value at $x = 5$.
  • Find the $y$ value at a point very close to $x = 5$ (e.g., $5.001$).
  • Label the points and plug them into the slope equation: $\frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$.
  • Calculating this will provide you the slope of the tangent line between these two points $0.001$ apart.

Topic 1.4: Polynomial Functions

• Polynomial Function: A function that can be written in the form:
  • $a<em>n x^n + a</em>{n-1} x^{n-1} + … + a<em>1 x + a</em>0$
  • Where $a_n$ is the coefficient and $n$ is the degree.
  • The degree gets smaller throughout the equation.
• Polynomial Requirements:
  • No negative degrees.
  • No imaginary coefficients.
  • No division within its equation.
• Common Polynomials:
  • Linear
  • Quadratic
  • Cubic
  • Quartic
• Local/Relative Maximum or Minimum: A maximum or minimum within a specific interval of the function.
• Global/Absolute Maximum: The greatest of all local maximums.
• Global/Absolute Minimum: The least of all local minimums.
• Infinite functions tending towards +∞ or -∞ do not count as maximums or minimums.
• Even Degree Polynomials: Always have a global/absolute maximum or minimum.
• Between Two Zeros: There is always a local maximum or minimum.
• Points of Inflection: Input values where the rate of change changes from increasing to decreasing, or vice versa; where the concavity changes.

Topic 1.5: Polynomials - Multiplicity, Degree, Even/Odd

• Zeros of a Function: Values of $x$ when $y=0$.
  • Also called x-intercepts or roots.
  • Can be real or imaginary.
  • Real zeros are shown on a graph.
  • Imaginary zeros require the imaginary number $i = \sqrt{-1}$.
  • Solutions with $i$ are imaginary.
  • Real zeros are linear factors, and imaginary zeros are complex zeros.
• Degree of a Function: The highest exponent value.
  • Can be calculated from a table by calculating successive differences until the numbers are the same; count the number of differences calculated.
• The degree of the equation dictates how many zeros (real or imaginary) the function has.
• Conjugate Rule: If $a + bi$ is a solution, then $a - bi$ is also a zero.
• Given an equation in intercept form:
  • Calculating Zeros: Set each factor equal to zero and solve.
• Graphing Zeros:
  • Examine the exponent next to each factor in intercept form.
  • Odd Exponent/Multiplicity: The line passes through the zero (odd multiplicity).
  • Even Exponent/Multiplicity: The line bounces off the zero instead of passing through it (even multiplicity).
• Even Function:
  • Satisfies $f(-x) = f(x)$.
  • Graph looks the same when reflected across the y-axis.
• Odd Function:
  • Satisfies $f(-x) = -f(x)$.
  • Graph looks the same when rotated 180 degrees across the origin.

Topic 1.6: End Behavior and Polynomial Functions

• End behavior describes how the graph ends on the left and right sides.
• Notation: Limit notation.
• Example:
  • Examine where $x$ ends.
  • Right: $x$ increases without bound (approaches positive infinity).
  • Left: $x$ decreases without bound (approaches negative infinity).
  • Write: $\lim<em>{x \to \infty}$ and $\lim</em>{x \to -\infty}$
  • Add $f(x) =$ to symbolize the y value.
  • Examine where $y$ goes as $x$ approaches infinity and negative infinity.
• Finding End Behavior from Equations - Trick:
  • Find the degree of the equation (odd or even).
  • Determine if the leading coefficient is positive or negative.
  • Refer to the a table relating degree parity and coefficients.
  • The start of end behavior equations is always: $\lim<em>{x \to \infty}$ and $\lim</em>{x \to -\infty}$
  • Table shows what $y$ equals under different scenarios.
• End behavior as it comes in from the left of the graph = $\lim_{x \to -\infty}$
• End behavior as it comes in from the right of the graph = $\lim_{x \to \infty}$

Topic 1.7: End Behavior in Rational Functions

• Rational functions are two polynomials divided ($\frac{polynomial}{polynomial}$).
• This division causes vertical and horizontal asymptotes to appear.
  • Asymptotes: Invisible lines that the graph approaches but never touches.
• End behavior in rational functions focuses on what $y$ does when $x$ approaches positive and negative infinity.
• Finding end behavior from a rational function equation:
  • 1. Bottom Heavy: Higher degree in the denominator.
    • $\lim_{x \to \pm\infty} f(x) = 0$
    • Horizontal asymptote at $y = 0$.
  • 2. Same Heavy: Numerator and denominator have the same degree.
    • $\lim_{x \to \pm\infty} f(x) = \frac{leading \ coefficient}{leading \ coefficient}$
    • Horizontal asymptote at this ratio.
  • 3. Top Heavy: Higher degree in the numerator.
    • If the degree is even:
      • $\lim_{x \to \infty} f(x) = \infty$
      • $\lim_{x \to -\infty} f(x) = \infty$
    • If the degree is odd, then swap the limits.
    • No horizontal asymptotes, but slant or oblique asymptotes exist.
      • Solve for these by using polynomial long division; the quotient is the slant/oblique asymptote.

Topic 1.8: Real Zeros of Rational Functions

• To solve for real zeros, set the numerator of the fraction equal to zero and solve.
• Also, set the denominator equal to zero and solve.
• If zeros match between the numerator and denominator, it is a hole, not a zero.
• Any remaining zeros that don't match in the numerator are the real zeros of the function.

Topic 1.9: Vertical Asymptotes in Rational Functions

• Set both the numerator and denominator equal to zero.
• Cross out any zeros that match between the numerator and the denominator (these are holes).
• The zeros remaining are your vertical asymptotes.
• Asymptotes are invisible lines that the graph approaches but never touches.
• Limit notation:
  • Parent Function: $\lim_{x \to asymptote^-} = \infty$
  • $\lim_{x \to asymptote^+} = -\infty$
• Vertical asymptotes prevent any number on the domain and range from being in that specific location.

Topic 1.10: Holes in Rational Functions

• A hole occurs when the numerator and denominator have a common factor.
• A hole on the graph is indicated with an open circle.
• If a hole exists at the point $(c, l)$, then $\lim_{x \to c} f(x) = l$ because nothing exists in that hole.
• Holes affect the domain and range of a function.

Topic 1.11: Building Polynomial Functions

• Given roots of a function, build a function by putting them in parentheses and multiplying each factor.
• Polynomial Long Division: Used to shrink a polynomial down when it has too many values in its expression.
  • Multiply then subtract and rinse and repeat till you are done.
  • Goal is to make it so whatever you put on top multiplies by the divisor to get whatever the first degree of the dividend is.
  • Remainder is added at the end of the quotient over the divisor.
• Binomial Theorem Using Pascal's Triangle: Shortcut way to foil out $\left(x + 5\right)^5$ without actually using distribution.
  • Pascal's triangle is a diagram of numbers where each number is the sum of the numbers above it.
  • Pattern of exponents rise and go down.

Topic 1.12: Transformations

• Transformations split into two groups:
  • Additive
  • Multiplicative
• Additive Transformations:
  • Vertical Translation: $g(x) = f(x) + k$
    • Moves the graph up or down.
  • Horizontal Translation: $g(x) = f(x + k)$
    • Moves graph of $f$ by -k units (left or right).
    • Remember that in the parentheses it's always the opposite.
• Multiplicative Transformations:
  • Vertical Dilation: $g(x) = a \cdot f(x)$, where $a \neq 0$
    • a > 1 makes the graph narrower.
    • 0 < a < 1 makes the graph wider.
    • If $a$ is negative, it reflects the graph over the x-axis.
  • Horizontal Dilation: $g(x) = f(bx)$, where $b \neq 0$
    • Horizontal dilation of $f$ by a factor of $\frac{1}{b}$.
    • If $b$ is negative, it reflects over the y-axis.
• Combined transformations affect the domain and range.

Topic 1.13: Predicting Function Models by Context Clues

• Linear Function: Constant rate of change.
• Quadratic Functions:
  • Rate of change shifting.
  • Associated with a function that has one distinct minimum or maximum.
  • Geometric contexts involving area or two dimensions.
• Cubic Functions: Geometric contexts that involve volume or three dimensions.
• Piecewise Function: Datasets/scenarios with different characteristics over different intervals.
• Modeling Questions:
  • Read the entire question and understand what it is asking and saying.
  • Model each answer to the real-world scenario given in the problem.
• Domain and range restrictions might exist (e.g., can't have 0.5 people).

Topic 1.14: Constructing Function Models

• Linear regression.
• Quadratic regression.
• Cubic regression.
• Quartic regression.
• Exponential regression.
• Determine the function model to use by using the graphing calculator and using the proper statistic test, find an r value close to one.
• Store regression; VARS, Y-VARS, FUNCTION, Y1.
• Graph and see what the graph will look like.

Topic 2.1: Sequences

• Sequence: A list of numbers.
  • Arithmetic: Numbers have a common rate of change with a common difference.
  • Geometric: Increases more and more, and multiplying by the same value each time.
• Equations:
  • Arithmetic:
    • $a<em>n = a</em>0 + dn$
      • $a_n$: Value of the term you are finding.
      • $a_0$: First term in the sequence.
      • $d$: Common difference
      • $n$: Position number you are trying to find.
    • $a<em>n = a</em>k + d(n - k)$
      • $a_k$: Number of the already known term.
      • $k$: Position number of the known term.
  • Geometric:
    • $g<em>n = g</em>0 \cdot r^n$
      • $g_n$: Term you are finding
      • $g_0$: First term of the sequence.
      • $r$: Common ratio, proportional change.
      • $n$: Position number of the term you are trying to find.
    • $g<em>n = g</em>k \cdot r^{n-k}$
      • $g_k$: The term you know.
      • $k$: Position number of the term you know.

Topic 2.2: Sequences Clarification & Exponential Functions

• Clarification on Sequence Formula Zero Term:
  • The zero term is the initial term AKA technically the term before the first term.
• Arithmetic Sequences: Linear Functions
  • Arithmetic Equation: $a<em>n = a</em>0 + dn$ is functionally equivalent to $y = mx + b$.
  • The other equation, $a<em>n = a</em>k + d(n - k)$, could be expressed as another way of writing linear equations being $f(x) = y<em>I + m(x - x</em>I)$, where you include the point $(x<em>I, y</em>I)$
• Geometric Sequences: Exponential Functions
  • Skeleton Equation: $f(x) = ab^x$
  • The second geometric sequence equation can be transformed into an exponential function written as $f(x) = y<em>I \cdot r^{x -x</em>I}$, where it includes the point $(x<em>I, y</em>I)$
• Linear Functions: Output values change at a constant rate based on addition.
• Exponential Functions: Output values change at a proportional rate based on multiplication.

Topic 2.3: Exponential Functions & Their Properties

• Skeleton Equation: $f(x) = ab^x$, where $a$ is the initial value and $b$ is the base.
• Rules of Exponential Functions:
  • $a \neq 0$
  • b > 0
  • $b \neq 1$
• Exponential Growth: Growth vs. Decay.
  • a > 0 and b > 1 (looks like line curves upwards).
  • Exponential Decay: a > 0 and 0 < b < 1 (looks like line curves downwards).
• All exponential functions have a domain of all real numbers.
• An exponential function is either always increasing or always decreasing.
• Only concave up or down and never changes, meaning exponential functions have no points of inflection.
• Each exponential function does not have a parent function but a form called the parent function $b^x$, where b > 1 for growth and 0 < b < 1 for decay.
• Parent Functions Properties:
  • There will always be a point at $(0, 1)$ because anything to the power of zero is one.
• There is a horizontal asymptote at $y = 0$.
  • The limit as $x$ approaches negative infinity of any growth parent function would be zero.
  • As $x$ approaches positive infinity, all growth parent functions would approach positive infinity.
• End behavior of any parent exponential decay functions would simply be swapped from the growths.

Topic 2.4: Rules of Exponents

• Product Property: $b^m \cdot b^n = b^{m+n}$
  • If you multiply two values with an exponent that have the same base or b, then you are really just adding their exponents
  • On a graph, this is a horizontal dilation to the graph that shifts to the right.
  • $\mapsto$ $y = 2^x$ becomes $y = 2^{(x - 1)}$
• Power Property: $(b^m)^n = b^{mn}$
  • If you have double exponents, you are simply multiplying the exponents together.
  • On a graph, this is either stretching or shrinking value to the graph, meaning it is the horizontal dilation.
• Negative Exponent Property: $b^{-n} = \frac{1}{b^n}$
  • If you ever have a negative exponent, it is simply equal to one over the original term removing the negative sign on the exponent.
• Exponent Root Property: $b^{\frac{1}{k}} = \sqrt[k]{b}$
  • If you ever have a power that is one over something, it's really asking you to do what the denominator's root of the function is.

Topic 2.5: Building Exponential Functions from Scenarios

• An exponential function is a best fit for a scenario if it is based on multiplication.
• Constructing an exponential function:
  • Only need two points or input-output value pairs to derive an exponential to fit the model.
  • Do this by solving a system of equations.
  • Use the transformations to tweak the functions to what the questions are asking.
• Exponential functions represent interest and compound interest in real life.
• E is a massively long number that is approximately equal to 2.718.
  • E is the base of a natural exponential function that is used to model continuous growth or decay in real life scenarios.
  • Allows us to model processes that change at a rate proportional to their current value.
  • Can be modeled with the EXP regression.

Topic 2.6: Function Modeling and Residuals

• Making Equations for Data Sets:
  • Linear: $y = mx + b$
    • $b$ = whatever $y$ equals when $x = 0$
    • $m$ = slope.
  • Quadratic: $y = a(x - b)^2 + c$
    • $c$ = vertical shift.
    • $b$ = horizontal shift.
    • Pick a coordinate point and plug and chug to find the last unknown, a.
  • Exponential: $y = ab^x$
    • $a$ = whatever $y$ equals when $x = 0$
    • $b$ = the rate at which the data is being multiplied.
• Calculator Regression:
  • You'll likely be given a dataset and asked to find an equation that best fits it, which, of course, you would find with a regression.
  • Once you've determined the regression equation, you might be given residuals. A residual is just the difference between the actual data point and the value predicted by your model. So if your model says the point should be there, but the actual data is a little higher or lower, the residual is the vertical distance between those two points. A model is considered appropriate if the residual plot, which is a graph of all the residuals, appears without a pattern. In other words, if the errors are random, then your model is a good fit for your data. If the residual plot shows a pattern, it means your model isn't fully capturing the behavior of the data, meaning the goal is to see randomness. The difference between the predicted and actual values is the error in the model.
    • Once you've determined the regression equation, you might be given residuals.
    • A residual is just the difference between the actual data point and the value predicted by your model.
    • A model is considered appropriate if the residual plot appears without a pattern; you want randomness.
  • Too high of an overestimate or underestimate of a dataset can be useful depending on the context.

Topic 2.7: Function Composition

• If given two functions and asked to find $f(g(x))$, simply substitute every instance of $x$ within $f(x)$ for $g(x)$. and solve.
• If asked to find $f(g(x))$ where $f(x) = x$, then the answer is $g(x)$.
• Remember that you can break a function down into two functions, and the original function would now become the result of g of h of x.
  • $f(x) = \sqrt(1 + x^2)$.
  • This can be broken down into two functions.
  • $g \cdot h$ would then rewrite function.
• Also, just so you know, g(h(x)) is the same as g \cdot h.

Topic 2.8: Inverse Functions

• An inverse is typically notated like instead of $f(x)$ being $f^{-1}(x)$.
• Find inverse by swapping the $x$ and the $y$.
• Take the parent cubic $y = x^3$.
  • Take it's first three point (1,1) (2,8) (3,27).
  • This shows how to typically solve for inverse function: swap x and y coordinates.
• For a function to have an inverse function, it must be one-to-one, meaning each output value is produced by exactly one input value.
• On the AP exam, be able to explain why an inverse exists or doesn't exist (think one-to-one function).
• Know a function is one to one if it passes the horizontal line test.
• The inverse function and the original function swap domain and ranges.

Topic 2.9: Logarithmic Expressions

• If you had the expression $2^x = 8$, you can of course infer that $x = 3$
  • To rearrange this into log form, we say log base two of eight. The answer to this is three.
  • $\log_b (c) = a$
  • If written in exponential expression it can be written $b^a = c$
• Two rules of logarithmic expressions:
  • $b$ has to be positive.
  • $b$ cannot be one.
• If you ever see a log with no base or b, it is known as a common log and the base is automatically 10 (written as $log(x)$.
• Logarithms bring a new scale to life.
  • Example: On a stand scale the units are 0, 1, 2, and so on. But logarithms bring a new scale to life and now look logarithmic.
• Math Alpha can be inputted on the calculator to solve logarithms with different bases.
  • Once this is habit to solve with, its great tool for exponential functions to solve for what the power will equal.

Topic 2.10: Logarithmic Functions Introduction

• Log Functions: A function expressed:
  • $a \log_b(x)$
  • where
  • $A$ cannot equal 0.
  • $B$ has to be positive
  • $B$ cannot be one.
• Well Log functions are inverse to exponentials as the coordinate points are swapped.
• Meaning, if an exponential function has a point at (t,s) the inverse log function will have a point at (s,t)
  Look at a log graph. Its a reflection of it's exponential counterpart flipped over the line $y = x$
• In theory if you don't understand logs, find the exponential equation of it, and then swap the coordinate points.
• A Log graph doesn't exist before a certain domain, just like exponential graphs.

Topic 2.11: Logarithmic Functions Properties

• Log functions are not so different from exponential functions
  • because they are inverses of one another
• The same rules of a parent function apply to a log graph
• The parent function is log base b of x, where b is between zero and one or greater than one.
• For all parent functions the domain will be any real number greater than zero [$x$ > 0]
  • Because of vertical asymptote at $x = 0$.
• The range would be all real numbers
• Log functions general format looks vertically asymptotic to $x equals 0$
• Since they are asymptotic the end behaviors are different
  • $x$ does not go to both positive and negative any longer
    • Instead $x$ goes to $positive infinity$ as the horizontal goes to positive. The vertical goes to 0
    • $The 1.12 video back in unit one will always to the log functions$