AP Precalculus Comprehensive Notes

Topic 1.1: Change in Tandem

This topic focuses on the relationship between input and output values in functions.

• Function Definition: A function is a mathematical relation where each input (x) maps to exactly one output (y).
• Input Values:
  • Also known as the domain of the function.
  • Referred to as the independent variable.
• Output Values:
  • Also known as the range of the function.
  • Referred to as the dependent variable.
• Increasing Function: As input values increase, output values also increase.
• Decreasing Function: As input values increase, output values decrease.

Graphs of Functions

• Definition: The graph displays input-output pairs, showing how values vary.
• Rate of Change: Equivalent to the slope of the graph.
• Concavity:
  • Concave Up: The rate of change is increasing; the graph resembles a U shape.
  • Concave Down: The rate of change is decreasing; the graph resembles an upside-down U shape.
• Zeros of a Function:
  • Occur where the graph intersects the x-axis.
  • The output value (y) is zero at these points.
  • The corresponding x-values are the zeros of the function.
• Equation: y = mx + b
  • x = input
  • y = output
  • m = slope (rate of change)
  • b = y-intercept (where the graph crosses the y-axis)

Topic 1.2: Rate of Changes

This topic explains how to calculate and describe rates of change.

• Rate of Change Definition: The slope of a graph.
• Slope Equation: \frac{y2 - y1}{x2 - x1}
• Finding Average Rate of Change:
  1. Given two points (x1, y1) and (x2, y2).
  2. Plug the points into the equation.
  3. Solve for the average rate of change.
• Using Average Rate of Change to Predict Points: Multiply a given value by the average rate of change.
• Positive Rate of Change: As one quantity increases or decreases, the other quantity does the same.
• Negative Rate of Change: As one quantity increases, the other decreases.

Topic 1.3: More on Rate of Change

This video expands on the concept of rate of change.

• Linear Function: The rate of change is constant across any interval (average rate of change is changing at a rate of zero).
• Quadratic Functions: The average rate of change changes at a linear rate.
  • Calculating the average rate of change over an interval determines the slope of the secant line.
  • This rate of change is not accurate to the quadratic function itself.
• Calculating Average Rate of Change at a Specific Point:
  1. Given a point (e.g., x = 5), find the y-value.
  2. Find the y-value at a point very close to it (e.g., 5.001).
  3. Use the slope equation with these two points.
  4. This calculates the slope of the tangent line between the two points.

Topic 1.4: Polynomial Functions

This topic explains the components and terminology of polynomial functions.

• Polynomial Function (Technical Form):

  • anx^n + a{n-1}x^{n-1} + … + a1x + a0
    • a_n is the coefficient.
    • n is the degree.

• Polynomial Requirements:

  • No negative degrees.
  • No imaginary coefficients.
  • No division within the equation.

• Common Polynomials: Linear, quadratic, cubic, and quartic.

Terminology

• Local (Relative) Maximum/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.
• Infinity: Functions going to +\infty or -\infty do not have maximums or minimums.
• Even Degree Polynomials: Always have an absolute maximum or minimum.
• Local Maximum/Minimums Between Zeros: Between two zeros, there will always be a local maximum or minimum.
• Points of Inflection: Occur where the rate of change changes from increasing to decreasing, or vice versa (where concavity changes).

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

This topic covers zeros, multiplicity, degree, and even/odd functions related to polynomials.

• Zeros of a Function:
  • Values of x when y = 0.
  • Also known as 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}.
  • Real zeros are also called linear factors, and imaginary zeros are called complex zeros.
• Degree of a Function:
  • The highest value of the exponent.
  • To calculate from a table, find successive differences from the y-side until the numbers are the same; the number of differences is the degree.
  • The degree of the equation also indicates the number of zeros (real or imaginary).
  • For a solution in the form a + bi, the conjugate a - bi is also a zero.
• Graphs and Intercept Form:
  • From intercept form, set each parenthesis to zero to calculate zeros.
  • The exponent next to the parenthesis indicates multiplicity.
    • Odd Multiplicity: The line passes through the zero on the graph.
    • Even Multiplicity: The line bounces off the zero on the graph.
• Even Functions:
  • Satisfy the property f(x) = f(-x).
  • The graph looks the same when reflected across the y-axis.
• Odd Functions:
  • Satisfy the property f(x) = -f(-x).
  • The graph looks the same when rotated 180° across the origin.

Topic 1.6: End Behavior and Polynomial Functions

This topic explains end behavior and how to determine it in polynomial functions.

• End Behavior Definition: Describes how the graph ends on the left and right sides.
• Limit Notation:
  • As x increases without bound, x \rightarrow \infty.
  • As x decreases without bound, x \rightarrow -\infty.
  • Examine what y (or f(x)) is doing as x does these things.
• Example:
  • If, as x \rightarrow \infty, y \rightarrow \infty, then the limit as x approaches infinity of f(x) equals infinity.
  • If, as x \rightarrow -\infty, y \rightarrow \infty, then the limit as x approaches negative infinity of f(x) equals infinity.
  • Minus in the exponent of the end behavior means coming in from the left; plus means from the right.
• Finding End Behavior from an Equation:
  • Find the degree of the equation and whether the leading coefficient is positive or negative.
  • Use a table to determine what y equals under these scenarios.

End Behavior Table

• Odd Degree & Positive Leading Coefficient:
  • \lim_{x \to \infty} f(x) = \infty
  • \lim_{x \to -\infty} f(x) = -\infty
• Odd Degree & Negative Leading Coefficient:
  • \lim_{x \to \infty} f(x) = -\infty
  • \lim_{x \to -\infty} f(x) = \infty
• Even Degree & Positive Leading Coefficient:
  • \lim_{x \to \infty} f(x) = \infty
  • \lim_{x \to -\infty} f(x) = \infty
• Even Degree & Negative Leading Coefficient:
  • \lim_{x \to \infty} f(x) = -\infty
  • \lim_{x \to -\infty} f(x) = -\infty

Topic 1.7: Rational Functions and End Behavior

This topic discusses rational functions and how to calculate end behavior.

• Rational Functions Definition: Two polynomials divided by one another.
• Vertical and Horizontal Asymptotes: Invisible lines that the graph approaches but never touches.
• End Behavior in Rational Functions: Focus on what y does as x approaches positive and negative infinity.
• Equation Rules:
  • Bottom Heavy: Higher degree in the denominator.
    • \lim_{x \to \pm \infty} f(x) = 0
    • Horizontal asymptote at y = 0.
  • Same Heavy: Same degree in numerator and denominator.
    • \lim_{x \to \pm \infty} f(x) = (ratio of leading coefficients)
    • Horizontal asymptote at this ratio.
  • Top Heavy: Higher degree in the numerator.
    • \lim_{x \to \infty} f(x) = \infty
    • \lim_{x \to -\infty} f(x) = -\infty or \infty (if the degree is odd; swap if the leading coefficient is negative).
    • No horizontal asymptotes: slant or oblique asymptotes exist.
      • Solve using polynomial long division.

Topic 1.8: Real Zeros of Rational Functions

This topic explains how to find real zeros of rational functions.

• Finding Real Zeros: Set the numerator equal to zero and solve.
• Denominator Consideration:
  • Set the denominator equal to zero and solve.
  • Any matching zeros between the numerator and denominator are holes.
  • Remaining zeros in the numerator are real zeros of the function.

Topic 1.9: Vertical Asymptotes in Rational Functions

This topic explains how to identify vertical asymptotes in rational functions.

• Finding Vertical Asymptotes:
  • Set the numerator and denominator equal to zero and solve.
  • Cross out/remove any matching zeros between the numerator and the denominator as those are holes.
  • Any remaining zeros are the vertical asymptotes.
• Asymptote Definition: An invisible line that a graph approaches but never reaches, all the way to positive or negative infinity.
• Limit Notation:
  • Approaching vertical asymptotes from the left or right in the parent rational function would be +\infty and -\infty.
  • This also works with horizontal asymptotes.

Topic 1.10: Holes in Rational Functions

This topic focuses on identifying and understanding holes in rational functions.

• Hole Definition: Occurs when there is a common factor between the numerator and denominator (once solved).
• Graphical Representation: Indicated by an open circle on the graph.
• Effect: A point where nothing exists on the graph.
• Limit Notation at a Hole: If a hole exists at the point (C, L), then \lim_{x \to C} f(x) = L.
• Domain and Range: Holes will affect the domain and range of the function.

Topic 1.11: Building Functions

This topic covers building functions from roots, polynomial long division, and the binomial theorem (Pascal's Triangle).

• Building a Function from Roots: Put the roots in parentheses and multiply each factor.
• Polynomial Long Division: Used to shrink a polynomial down.
  • Goal: Make whatever is on top multiplied by the divisor get whatever the first degree of the dividend is.
  • Procedure: Multiply, then subtract, and repeat until done.
  • Any remainder is added to the quotient over the divisor.
• Binomial Theorem Using Pascal's Triangle: A shortcut, e.g., solving (x + 5)^5.
  • Pascal's triangle is a diagram where each number is the sum of the numbers above it.

Topic 1.12: Transformations

This topic covers additive and multiplicative transformations.

• Additive Transformations:
  • Vertical Translation: g(x) = f(x) + k (moves the graph up or down).
  • Horizontal Translation: g(x) = f(x - k) (moves the graph left or right by negative k units).
• Multiplicative Transformations:
  • Vertical Dilation: g(x) = a f(x)
    • When a \neq 0, the higher the number, the more closed the graph; the lower the number, the more open it gets
    • If a is negative, it reflects the graph over the x-axis.
  • Horizontal Dilation: g(x) = f(bx)
    • When b \neq 0, by a factor of 1/b.
    • If b is negative, the result is a reflection over the y-axis.
• All transformations can be combined and used in one equation.

Topic 1.13 & 1.14: Predicting and Constructing Function Models

These topics cover function modeling using context clues and constructing those functions.

• Linear Function: Models data with a constant rate of change.
• Quadratic Functions:
  • Rate of change shifts.
  • Typically associated with a function that has one minimum or maximum.
  • Geometric contexts involving area or two dimensions.
• Cubic Functions: Geometric contexts involving volume or three dimensions.
• Piecewise Function: Different characteristics over different intervals.
• Underlying Assumptions: Read the problem entirely to fully understand what it is asking.
• Real-World Scenarios: Be aware of domain and range restrictions.

Calculator Method

1. Press the stat button, then edit.
2. Input a set of data to L1 (x values) and L2 (y values).
3. Press stat, then calc.
4. Run regressions (linear, quadratic, cubic, and quartic).
5. The regression with the R value closest to one is the best-fitting model.
6. To store the regression, go to vars, then y vars, function, and store it in Y1.

Topic 1.13 & 1.14: Rational Functions on Calculator

• Rational functions can be manually put in the calculator.
• You can find the value of y when x is equal to something.

Topic 2.1: Sequences

This covers arithmetic and geometric sequences and their formulas.

• Sequence Definition: A list of numbers.
• Arithmetic Sequence: A linear function with a common rate of change (common difference).
• Geometric Sequence: Increases more and more due to a common proportional change (multiplied by a number).

Arithmetic Sequence Equations

• an = a0 + dn
  • a_n = value of the term you're finding.
  • a_0 = the first term in the sequence.
  • d = common difference.
  • n = the position number you are trying to find.

Not knowing the first term

• an = ak + d(n - k)
  • a_k = already known term.
  • k = position number of the known term.

Geometric Sequence Equations

• gn = g0 * r^n
  • g_n = term you are finding.
  • g_0 = the first term in the sequence.
  • r = common ratio of proportional change.
  • n = position number of the term you're trying to find.

Not knowing the first term

• gn = gk * r^(n - k)
  • g_k is the term you know
  • k = the position number of the term you know

Topic 2.2: Clarification on Sequences and Introduction to Exponential Functions

• Zero term is the initial term, but it is technically the term before the first term.
• Arithmetic sequences are linear functions.
• an = a0 + dn is really just the equation y = b + mx.
• an = ak + d(n - k) could be expressed as f(x) = yi + m(x - xi).
• Linear functions have output values changing at a constant rate from addition.
• Exponential functions have output values changing at a proportional rate from multiplication.

Topic 2.3: Exponential Functions and Their Properties

This topic focuses on exponential functions and their properties.

• Skeleton Equation: f(x) = a * b^x
  • a = initial value
  • b = base
• Rules of Exponential Function:
  • a cannot equal zero.
  • b must always be positive.
  • b can never be one.
• Exponential Growth: a > 0 and b > 1.
• Exponential Decay: a > 0 and 0 < b < 1.
• Domain of All Exponential Functions: All real numbers.
• Concavity: Either always concave up or down (no points of inflection).
• Parent Function: b^x, where b > 1 for growth or 0 < b < 1 for decay.
• Fun Things About Parent Functions:
  • There will always be a point at (0, 1).
  • There is a horizontal asymptote at y = 0.
  • \lim_{x \to -\infty} of any growth parent function is zero.
  • \lim_{x \to \infty} of any growth parent function is \infty.
  • End behavior of any parent exponential decay functions would simply be swapped from the growths

Topic 2.4: Rules of Exponents

This topic discusses different rules dealing with exponents.

• Product Property: b^m * b^n = b^(m + n)
  • Adding or subtracting anything from x is a horizontal translation of the graph.
• Power Property: (b^m)^n = b^{mn}
  • Acts as the stretch or shrink value to the graph (horizontal dilation).
  • To sketch a graph that has dilations, simply make a table and graph that information to make it easiest.
• Negative Exponent Property: b^{-n} = \frac{1}{b^n}
• Exponent Root Property: b^{\frac{1}{k}} = \sqrt[k]{b}

Topic 2.5: Building Exponential Functions

This covers building exponential functions from real life scenarios.

• Tip: look for a multiplication exponential function.
• Given two points to derive an exponential function from the model by solving a system of equations.
• B is the rate of growth for exponential functions written to represent exponential interest and compound interest in real life.
• e = 2.718, this is the base of a natural exponential function that is used to model continuous growth or decay in real-life scenarios.
• On a calculator, in the regression you have an exp regression which can be run to see if the given data is exponential.

Topic 2.6: Function Modeling and Residuals

Skeleton equations overview for Linear, QAudratic, and Exponential functions

• If the data is linear, y = mx + b \to whatever y = 0 = b , and the slope is that formula
• If the data is quadratic, y = a(x - b)^2 = c \to the vertical shift = c, the vertical translation = b, and using a point with algebra to solve for a.
• f the data is exponential, y= a*b^x \to whatever y is when x = 0 = a, and rate that the data is being multiplied by = b
• A residual is the vertical distance between the actual data point, and the model says it should be.
  • A model is apropriate if the residual plot ( the garaph the residuals), appeards without a pattern.
  • With the goal of seeting randomness, with error between predicted, and actual values with an over/underestimate depends.

Topic 2.7: Function Composition

• Given two function and are asked to solve the equation, you will substitue the instance of x within the (f(x)) function to the (g(x)) function and solve.
• Remember that you can break it down into two functions, and the original function would now become the result of g(h(x)),
  • Do not get the two equations mixed up.

Topic 2.8: Inverse Functions

These are weird man, typically notated. f(x) \to f^{-1}(x).
To understand lets take a paretn Cubic fuction y = x^3 we would know that the fist three points on the graph would be (1,1), (2,8), (3,27), the inverse of this fuction would have tehse points, but simply swapping the X and Y, So it would be (1,1), (8,2), (27,3).

• To find the inverse function equation, you would take to equal y = x^3 \to swap the x and the y and solve it for the inverse fuction.
• For a function to have an inverese function, it must be one -to-one. meaning each output value is producesd by Exactly one input value.
• On a graph you Know a function is one to ome if it passess the horizonal line test- meaning if it intersepts the graph once- then itis one tonone,
  • Any more times then one and its not onne to one.
• Last thing about inerese fuctions- the inerese and original swao domaina nd ranges-
  • So the originals domain- eecoems the inverseses range, and originals range,, ecoems the insereses domain.

Topic 2.9: Intro to Logarithms

• These people dont enjoy, even teachers dont, is not bad to how u THINK ,U JUST NEED TO LEARN HOW THEY WORK!
• 2^x = 8 u can infrer that to equal 3
• to rearange this into log form, log28 = 3 to rearange, logbc = a, where b ^a = c.
• the Two rule sof logarthmic expressioons- b has ot be ooisitve, and b cant be one!
• also if you see a log with no base- you know the base izutomaticly equals 10!
• Logairthms bring a scale into light
  • Sdtard scale units migth be 0,1, 2 and so on!
  • logarhitmic with a ten as the base- the units will be 10^0,10^1,10^2, and o son!
• goinog over to calc- u cna imout this in to math alpham 1 and u cna input base and answer
• now cna habitually use this solve of powers. and you some times need to solve to exoonatial fuctions. i mmean heck look at pic 2.2 and u will use it there.