Pre Calculus Mastered

Explain natural numbers, integers, rational and irrational numbers, real numbers, imaginary numbers, and complex numbers, with symbols N, Z, Q, Q with a hat, and I, plus examples like pi.

Explore the coordinate plane, including the x and y axes, the origin, and quadrants, and how points are located by their x and y coordinates.

Learn to evaluate a function by substituting values into the placeholder and applying the rule to generate outputs, with notes on domain, range, and independent versus dependent variables.

Define the domain as the set of inputs that keep a function defined. Show examples: f(x)=x+1 accepts all real numbers, 1/(1-x) excludes 1, and 1/(x^2 - x - 2) excludes -1 and 2.

Discover how to find a function's domain, including rational functions by setting the denominator to zero and excluding those values, with notes on non-rational and square-root domains.

Determine the domain of root functions by ensuring the radicand is nonnegative for even roots; odd roots yield all real numbers as domain, including nine minus x^2 and 2x+9.

Explore the range of a function as the set of outputs produced from inputs, contrasted with the domain, with examples like f(x)=2x+1 and f(x)=x^2 showing nonnegative outputs.

Learn how graphs represent functions as pictures by plotting input-output pairs, linking domain and range, and graphing functions such as f(x)=x+1 and f(x)=x^2.

Explore piecewise functions by defining f(x) on separate intervals with distinct formulas, including domain, range, endpoints, holes, and inclusive or exclusive bounds.

Learn to extract information from a graph by locating x and reading the corresponding y as f(x), or finding x from a given y using lines.

Learn to find a graph's domain and range by identifying the x values fed into the function and the corresponding y values, including gaps in piecewise graphs.

Identify increasing and decreasing functions using the formal definitions: f(x1) < f(x2) when x1 < x2 on an interval, and analyze the graph with a polynomial example.

Learn to perform horizontal and vertical shifts on graphs, such as shifting f(x)=x^2 to (x-1)^2+4 (right 1, up 4) and shifting f(x)=sqrt(x) to sqrt(x-2) (right 2).

Learn how vertical stretching and shrinking modify graphs by multiplying f(x) by a constant greater than 1 to stretch, and by its reciprocal to shrink, using sin x and x^2.

Identify even and odd functions by symmetry about the y-axis and origin, using f(-x) tests. See x^2, x^4, x^6 as even; x^3 as odd; not all even powers guarantee evenness.

Explore how to add, subtract, multiply, and divide two functions, and evaluate the resulting function at given x values, illustrated with f(x)=x^2 and g(x)=x+1 and a graph example.

Combine two functions to find the domain by intersecting their domains when adding, subtracting, multiplying, or dividing, and exclude zeros in the denominator through worked examples.

Explore one-to-one functions by distinguishing inputs and outputs, ensuring unique outputs for each input, and apply the horizontal line test to identify non one-to-one cases.

Explore how the graph of an inverse function is obtained by swapping the domain and range and reflecting across the line y = x, with examples.

Discover how to find the range of a function algebraically by computing its inverse and using the inverse's domain as the original function's range, illustrated with examples.

Master polynomials by recognizing many-term structure, identifying variables, constants, and powers, and determining degree, leading coefficient, and leading term, while noting nonnegative powers and restrictions on division.

Analyze end behavior of polynomials by degree and leading coefficient to determine whether both ends rise, fall, or one end rises while the other falls.

Identify the real roots of polynomials by setting f(x) to zero and factoring to solve for x. These roots are the x-axis intercepts.

Use the intermediate value theorem for polynomials: if p(a) and p(b) have opposite signs, a root lies between a and b. Between consecutive zeros, the function remains positive or negative.

Explore local extrema, including local maximum and local minimum, on graphs and polynomials, and learn that a degree n polynomial has at most n minus one local extrema.

Master polynomial long division by dividing the highest-degree term by the divisor, multiplying, and subtracting to form the quotient and remainder until the remainder degree is less than the divisor.

Apply the factor theorem by substituting c into P(x); if P(c) equals zero, x minus c is a factor. Factor polynomials using division or synthetic division with a root.

Apply the rational zeros theorem to identify possible zeros of a polynomial as p/q, where p divides the constant term and q divides the leading coefficient.

Explore slant asymptotes in rational functions where the numerator's degree exceeds the denominator's by one, and learn how the quotient determines the asymptote and how the graph approaches it.

Learn to graph rational functions by factoring, locating x-intercepts and y-intercepts, identifying vertical, horizontal, and slant asymptotes, and analyzing graph behavior near these asymptotes.

Explore exponential functions, where f(x) = a^x with a > 0 and a ≠ 1, and where the base is a number and the exponent is a real variable.

Plot graph of exponential functions, noting the y-intercept at 1 and horizontal asymptote y=0. Compare bases above 1 with those between 0 and 1, and solve f(2)=25 for a.

Explore the natural exponential function with base e, learn that e is an irrational number about 2.718, evaluate f(x)=e^x, and examine how scaling or negating x shapes the graph.

Learn to solve exponential equations by making bases the same and equating the exponents. Practice with examples like 3^x=3^1, 8^x=8^{10}, 3^x=27, and 4^{x+1}=1/64 to find x.

Explore how interest grows with simple, compound, and continuously compounded methods using exponential functions, and learn key formulas like a = p(1 + r/n)^{nt} and a = p e^{rt}.

Explore the graph of logarithmic functions with base a, inverse of exponentials, noting domain (0, infinity), range all real numbers, an asymptote at x=0, and crossing the x-axis at (1,0).

Explore the natural logarithm as the inverse of the natural exponential, using ln (log base e) and its domain rules.

Master the core logarithm laws, including product, quotient, and power rules, apply the change of base formula, and simplify logs with base 10 conventions for clear problem solving.

Explore the fundamentals of trigonometry through the unit circle, with upcoming sections on the right angle triangle approach and proofs and identities.

Learn the unit circle, centered at the origin with radius one, via x^2 + y^2 = 1, and how to shift circles using (x-h)^2+(y-k)^2=r^2 while checking points by substitution.

Explore terminal points on the unit circle, determine their coordinates (x,y) for angles like π/6, π/4, π/3, and 2π, and master positive and negative rotations.

Define the six trig functions from the unit circle using the terminal point: sine is y, cosine is x, tangent is y over x; reciprocals yield cosecant, secant, and cotangent.

Evaluate trigonometric functions on the unit circle by using terminal points, quadrant signs, and reference angles; apply the add sugar to coffee memory aid to determine positivity and negativity.

explore sine and cosine graphs, applying amplitude and period changes, vertical shifts, and phase shifts to graph transformations, including left and right shifts and reflections across the x-axis.

Explore tangent and cotangent graphs and their transformations, including period pi, vertical asymptotes, and shifting with k, horizontal shifts, and scaling; learn to graph one period.

Explore secant and cosecant graphs as reciprocals of cosine and sine, learn their vertical asymptotes and period 2π, and how shifting the graphs yields one complete period.

Identify the domain and range of sine, cosine, and tangent; note tangent has vertical asymptotes, and transform ranges via inequalities, e.g., 3 sin(5x) - 2 yields [-5, 1].

Explore the domain and range of sec, csc, cot, and tangent by analyzing their graphs, vertical asymptotes, and how input and constant changes alter domain and range.

Examine the domain and range of cotangent, cosecant, and secant, focusing on vertical asymptotes and transformed forms like 5x - π + 8 and x - 2π.