Free step-by-step math lessons with guided notes to help you truly understand math.
Welcome to Understand The Math, where you'll find clear explanations, worked examples, and free guided notes designed to help you build confidence and succeed in math.
Whether you're studying algebra, trigonometry, precalculus, calculus, linear algebra, or differential equations, you'll find lessons that focus on building real understanding, not just memorizing formulas.
Created by Dr. Cheryl Hile, a former Penn State mathematics professor with more than 25 years of university teaching experience and a Ph.D. in Engineering Science and Applied Mathematics.
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Graphing exponential functions becomes much easier when you understand how transformations affect the graph!
Exponential functions can represent growth or decay, and their graphs can be shifted, stretched, compressed, or reflected.
These notes cover the general form of an exponential function, how to identify transformations, and how to find the horizontal asymptote.
You'll also find a step-by-step example showing how to graph y = 3 · 2^(x+1) − 2.
The key is to identify the transformations before choosing points to plot!
Save these notes for your next algebra or precalculus exam!
#ExponentialFunctions #GraphingFunctions #Algebra2 #Precalculus #MathHelp #LearnMath
3 days ago | [YT] | 8
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Optimization problems are an important application of derivatives in Calculus 1!
The goal is to find the maximum or minimum value of a quantity, often subject to certain constraints.
These notes walk through a step-by-step approach to solving optimization problems, followed by an example where the goal is to find two numbers whose difference is 100 and whose product is a minimum.
The key is knowing how to set up the problem before taking the derivative!
Save these notes for your next calculus exam!
#Calculus1 #OptimizationProblems #CalculusHelp #Derivatives #LearnCalculus
5 days ago | [YT] | 4
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📘 **Linear Algebra: Subspaces Cheat Sheet**
What makes a subset of a vector space a subspace?
To be a subspace, a subset must satisfy three conditions:
1. Contain the zero vector
2. Be closed under addition
3. Be closed under scalar multiplication
This cheat sheet summarizes the definition and includes examples of subspaces in ℝⁿ, matrix spaces, and polynomial spaces.
📌 Save this for your next Linear Algebra exam!
🔗 Find free Linear Algebra video lessons and guided notes at www.understandthemath.com/linear-algebra
#LinearAlgebra #Subspaces #VectorSpaces
1 week ago | [YT] | 5
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Understand The Math
Need help finding asymptotes of rational functions?
Learn how to find vertical, horizontal, and slant asymptotes using clear steps and worked examples.
A helpful review for Precalculus students studying rational functions!
Watch the full lesson: https://youtu.be/ostZlcWXu_0
1 week ago | [YT] | 4
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Understand The Math
Working on second-order differential equations?
Learn how to solve homogeneous linear differential equations with constant coefficients, including initial value problems, boundary value problems, and higher-order equations.
Follow along with step-by-step examples and free guided notes!
Watch the full lesson: https://youtu.be/9H6IfrGLJpE
2 weeks ago | [YT] | 4
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Understand The Math
Learning how to find the inverse of a matrix in Linear Algebra?
This lesson explains how to use row operations on an augmented matrix [A | I] to find A⁻¹, with step-by-step examples.
Free guided notes are available to help you follow along!
Watch the full lesson: https://youtu.be/ZmMNp3Je73A
2 weeks ago | [YT] | 3
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Learning integration using partial fractions in Calculus 2?
This lesson walks through partial fraction decomposition with distinct linear factors, repeated linear factors, irreducible quadratic factors, and improper rational fractions.
Learn the methods with step-by-step explanations and worked examples!
Watch the full lesson: https://youtu.be/P20HCNDijSo
3 weeks ago | [YT] | 4
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Learning implicit differentiation in Calculus 1?
This step-by-step lesson explains the difference between explicit and implicit functions, how to use implicit differentiation to find derivatives, and how to work through several examples.
Free guided notes are also available!
Watch the full lesson: https://youtu.be/OIBaSu3vkPQ
3 weeks ago | [YT] | 0
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Do you remember the Product Rule?
The Product Rule is used to find the derivative of a product of two functions. In this short, I explain what the Product Rule is and work through a quick example showing exactly how to use it.
If you’re learning derivatives in Calculus 1 or AP Calculus AB, this is an important differentiation rule to know.
Watch the short here: youtube.com/shorts/jkFciKJuVl...
#ProductRule #Calculus #Derivatives
1 month ago | [YT] | 5
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Understand The Math
Need a quick Product Rule refresher? Here’s a cheat sheet with the formula and worked examples to help you remember how to differentiate the product of two functions.
The Product Rule:
f(x)g′(x) + g(x)f′(x)
Save this for your next Calculus 1 or AP Calculus AB review!
#ProductRule #Derivatives #Calculus #Calculus1 #APCalculus
1 month ago | [YT] | 4
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