01
Absolute value as distance
Why |x| makes a V, how to move that V around, and how to read solutions straight off the graph.
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All courses
MTH 161
Precalculus I
Presents topics in power, polynomial, rational, exponential, and logarithmic functions, and systems of equations and inequalities.
How this course is taught: Video lessons, interactive demos, and the free OpenStax textbook.
How to use this page
- Watch the lesson for the section you are on, in the player below.
- Experiment with the matching interactive demo to build intuition.
- Read and practice that section in the free OpenStax book linked in the outline.
Practice problems: each OpenStax Precalculus 2e section has exercises with answers in the back. The demo checks are for understanding, not a grade.
Video lessons
Watch the full course
Lessons
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Course outline
What we cover, and where to read it
Our lessons follow OpenStax Precalculus 2e, a free, peer-reviewed textbook. Each unit below lists its topics and links to the matching chapter so you can read alongside the videos.
Unit 1 · Chapter 1
Functions
- Function notation, domain and range
- Rates of change and graph behavior
- Composition and transformations
- Absolute value and inverse functions
Unit 2 · Chapter 2
Linear Functions
- Linear functions and their graphs
- Slope, intercepts, and equations of lines
- Modeling with linear functions
- Fitting linear models to data
Unit 3 · Chapter 3
Polynomial & Rational Functions
- Complex numbers and quadratic functions
- Power and polynomial functions and their graphs
- Dividing polynomials and finding zeros
- Rational functions
Unit 4 · Chapter 4
Exponential & Logarithmic Functions
- Exponential functions and their graphs
- Logarithmic functions and their graphs
- Properties of logarithms
- Solving exponential and logarithmic equations
Bonus · Chapter 11.6
The Binomial Theorem
- Expanding a binomial raised to a power without multiplying it all out
- Binomial coefficients and Pascal's triangle
Official grades, submissions, and course announcements live in Canvas. This site hosts public course materials, explanations, examples, and study resources.
Interactive demos
Explore the ideas, hands on
These tools let you move the pieces and watch the math respond. Each one walks through the idea first, then hands you the controls. Nothing here is graded, so experiment freely. Not sure how to study this? See Study Strategies. Back to the lessons
02
Slope as a rate of change
What slope really measures: how fast one quantity changes as another changes, read straight off the line.
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03
Transformations of functions
Pick a parent shape, then shift, stretch, and flip it with a·f(b(x − h)) + k.
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04
Exponential growth and decay
What happens when you multiply instead of add, step after step, and why the curve takes off.
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05
Logarithms: counting by multiplying
A logarithm is just a different way of counting. Instead of adding, you count how many times you multiply.
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06
How exponentials and logs relate
They undo each other. See why the two curves are mirror images across the line y = x.
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Quick reference
Notation, functions & formulas
The notation, function families, and algebra rules that run through Precalculus I, gathered in one place. When a video or demo uses a symbol you do not recognize, it is almost certainly defined here.
Key notation
| Symbol | What it means |
|---|---|
| f(x) | Function notation — the single output of the function f at input x. Read “f of x,” it does not mean f times x. |
| domain, range | The set of allowed inputs (domain) and the set of outputs they produce (range). |
| (f∘g)(x) | Composition — do g first, then feed the result into f: f(g(x)). |
| f−1(x) | Inverse function — undoes f. If f(a)=b, then f−1(b)=a. Not a reciprocal. |
| |x| | Absolute value — distance from zero, so it is never negative. |
| an | Power — the base a multiplied by itself n times. |
| loga x | Logarithm base a — the exponent you must raise a to in order to get x. |
| ln x, log x | Natural log (base e) and common log (base 10). |
| e | Euler’s number ≈ 2.71828, the natural base for growth and decay. |
Transformations of any parent function
Start from a parent shape f(x) — a line, parabola, √, |x|, ax — and every move comes from the same template: g(x) = a·f(b(x − h)) + k.
| Constant | Effect on the graph |
|---|---|
| a | Vertical stretch when |a|>1, shrink when |a|<1; reflects over the x-axis when a is negative. |
| b | Horizontal stretch/shrink by a factor of 1/|b|; reflects over the y-axis when b is negative. |
| h | Horizontal shift — right when h>0, left when h<0 (it moves opposite the sign inside). |
| k | Vertical shift — up when k>0, down when k<0. |
Formula sheet
Slope of a line
m = y2 − y1x2 − x1
Equations of a line
y = mx + b · y − y1 = m(x − x1)
Quadratic formula
x = −b ± √(b2 − 4ac)2a
Vertex of a parabola
x = −b2a
Exponent rules
aman = am+n (am)n = amn a−n = 1/an
Logarithm rules
log(xy) = log x + log y log(xn) = n log x
Change of base
loga x = ln xln a
Exponentials & logs are inverses
y = ax ⇔ x = loga y
Compound interest
A = P(1 + r/n)nt · A = Pert
Binomial theorem
(a+b)n = Σ nCk an−kbk
Glossary
Function
A rule that assigns exactly one output to every input. Passing the vertical-line test confirms a graph is a function.
Domain & range
All the inputs a function accepts, and all the outputs it can produce. Watch for division by zero and even roots of negatives.
Composition
Chaining functions so one’s output is the next one’s input, written (f∘g)(x) = f(g(x)).
Inverse function
The function that reverses another, swapping inputs and outputs. Its graph is the mirror image across y = x.
One-to-one
Every output comes from exactly one input (passes the horizontal-line test). Only one-to-one functions have inverses.
Asymptote
A line the graph approaches but never reaches, common for rational, exponential, and logarithmic functions.
End behavior
What a graph does as x runs far to the left and right, set by the leading term of a polynomial.
Extraneous solution
An answer that appears during algebra (often after squaring or with logs) but fails in the original equation. Always check.
Exponential growth & decay
Repeated multiplying instead of adding: growth when the base is > 1, decay when it is between 0 and 1.
Every topic here is covered in depth in the free OpenStax College Algebra / Precalculus textbook linked from the course outline above.
Turn it in
Your practice report
Every quick check above feeds this report: what you got right on the first try, and what you got right eventually. Download it and upload it to Canvas.
MTH practice report · bryantclass.com