# Math Lessons

Finally! A **free math website** that explains** math** in a simple way, and includes lots of examples, from Counting through Calculus!

Here are the topics that **She Loves Math** covers, as expanded below: **Basic Math**, **Pre-Algebra**, **Beginning Algebra**, **Intermediate Algebra**, **Advanced Algebra**, **Pre-Calculus**, **Trigonometry**, and **Calculus **(to be completed).

# Basic Math

**Decimals**

**Fractions**

**Introduction to Fractions****Adding Fractions, Simplifying Fractions****Subtracting Fractions****Multiplying Fractions****Dividing Fractions****Comparing Fractions****Fractions Used in Cooking**

**Metric System**

## More Practice:

# Pre-Algebra

**Percents, Ratios, and Proportions**** **

**Percentages and Percent Changes****Ratios and Proportions****Unit Multipliers****Using Percentages with Ratios**

**Negative Numbers and Absolute Value**

**Negative Numbers****Absolute Value****Adding and Subtracting Negative Numbers****Multiplying and Dividing Negative Numbers**

**Powers, Exponents, Radicals, and Scientific Notation**

**Order of Operations PEMDAS**** **

**Introduction to Statistics and Probability**

# **Beginning Algebra**

**Beginning Algebra**

**Introduction to Algebra **

**Types of Numbers and Algebraic Properties**

**Types of Numbers****Algebraic Properties****Summary of Algebraic Properties (Chart)****Proper Algebraic Notation**

**Solving Algebraic Equations**

**Solving Linear Equations****Solving Literal Equations (Transforming Formulas)****Algebraic Equations with Absolute Value**

**Linear Inequalities**

**Compound Inequalities****Absolute Value and Inequalities****Graphing Linear Inequalities with Two Variables**

**Word Problems in Algebra**

**Word Problems in Algebra**

**English to Math Translation****“Find the Numbers” Word Problems****Percent Word Problem****Percent Increase Word Problem****Ratio/Proportion Word Problems****Weighted Average Word Problem****Consecutive Integer Word Problem****Age Word Problem****Money (Coins) Word Problem****Mixture Word Problem****Percent Mixture Word Problem****Rate/Distance Word Problem****Profit Word Problem****Converting Repeating Decimal to Fraction Word Problem****Inequality Word Problems****Combined Variation Word Problems****Work Word Problems**(in Rational Functions and Equations)**Systems of Equations Word Problems****More Word Problems using Rational Functions**

**Coordinate System and Graphing Lines Including Inequalities**

**Coordinate System****Slope-Intercept Formula****Positive and Negative Slopes****Horizontal and Vertical Line****s****Graphing Lines****Obtaining an Equation for a Line****Parallel and Perpendicular Lines****Distance and Midpoint Formulas****Graphing Linear Inequalities with Two Variables**

**Direct, Inverse, Joint and Combined Variation**

**Direct or Proportional Variation****Direct Variation Word Problems****Inverse or Indirect Variation****Inverse Variation Word Problems****Joint Variation and Combined Variation****Combined Variation Word Problems**

**Introduction to the Graphing Display Calculator (GDC)**

# **Intermediate Algebra**

**Intermediate Algebra**

**Systems of Linear Equations and Word Problems**

**Introduction to Systems****Solving Systems by Graphing****Solving Systems with Substitution****Solving Systems with Linear Combination or Elimination****Types of Equations****Word Problems with Systems**

**Algebraic Functions**

**Algebraic Functions Versus Relations****Vertical Line Test****Domain and Range of Relations and Functions****Finding the Domain Algebraically**

**Scatter Plots, Correlation, and Regression**

**Exponents and Radicals in Algebra**

**Introducing Exponents and Roots (Radicals) with Variables****Properties of Exponents and Roots****Putting Exponents and Radicals in the Calculator****Rationalizing Radicals****Simplifying Exponential Expressions****Solving Exponential and Radical Equations****Solving Simple Radical Inequalities**

**Introduction to Multiplying Polynomials**

**Introduction to Quadratics**

**Solving Quadratics by Factoring and Completing the Square**

**Factoring Methods****Completing the Square (Square Root Method)****Completing the Square to get Vertex Form****Obtaining Quadratic Equations from a Graph or Points****Quadratics Review**

**Quadratic Inequalities**

**Graphing Quadratic Inequality Functions****Solving Quadratic Inequalities****Quadratic Inequality Real World Example**

**Quadratic Applications**

**Quadratic Projectile Problem****Quadratic Trajectory (Path) Problem****Optimization of Area Problem****Maximum Profit and Revenue Problems****Population Problem****Linear Increase/Decrease Problem****Pythagorean Theorem Quadratic Application****Quadratic Inequality Problem****Finding a Quadratic Equation from Points or a Graph**

## Solving Absolute Value Equations and Inequalities

**Solving Absolute Value Equations****Solving Absolute Value Inequalities****Graphs of Absolute Value Functions****Applications of Absolute Value Equations**

**Solving Radical Equations and Inequalities**

**Radical Function Graphs****Solving Radical Equations Algebraically****Solving Radical Equations Graphically****Solving Radical Inequalities Algebraically****Solving Radical Inequalities Graphically**

## Imaginary (Complex) Numbers

**Advanced Algebra**

**Advanced Functions: Compositions, Even and Odd, Increasing and Decreasing**

**Adding, Subtracting, Multiplying and Dividing Functions****Increasing, Decreasing and Constant Functions****Even and Odd Functions****Compositions of Functions (Composite Functions)**

**Inverses of Functions**

**Introduction to Inverses****Finding Inverses Graphically****Finding Inverses Algebraically****Finding Inverses with Restricted Domains****Using Compositions of Functions to Determine if Functions are Inverses**

**Parent Functions and Transformations**

**Basic Parent Functions****Generic Transformations of Functions****Mixed Transformations****Transformations in Function Notation****Writing Transformed Equations from Graphs****Transformations of Inverse Functions****Absolute Value Transformations****Applications of Parent Function Transformations**

## Piecewise Functions

**Introduction to Piecewise Functions****Evaluating Piecewise Functions****Graphing Piecewise Functions****How to Tell if a Piecewise Function is Continuous or Non-Continuous****Obtaining Equations from Piecewise Function Graphs****Absolute Value as a Piecewise Function****Transformations of Piecewise Functions****Piecewise Function Word Problems**

**The Matrix and Solving Systems with Matrices**

**Introduction to the Matrix****Adding and Subtracting Matrices****Multiplying Matrices****Matrices in the Graphing Calculator****Determinants, the Matrix Inverse, and the Identity Matrix****Solving Systems with Matrices****Solving Word Problems with Matrices****Cramer’s Rule****Number of Solutions when Solving Systems with Matrices**

**Introduction to Linear Programming**

**Review of Inequalities****Bounded and Unbounded Regions****Inequality Word Problem****Linear Programming Terms****Linear Programming Word Problems**

**Rational Functions and Equations**

**Graphing Rational Functions, including Asymptotes**

**Graphing Rational Functions, including Asymptotes**

**Revisiting Direct and Inverse Variation****Polynomial Long Division****Asymptotes of Rationals****Drawing Rational Graphs — General Rules****Rational Inequalities, including Absolute Values****Applications of Rational Functions**

**Graphing and Finding Roots of Polynomial Functions**

**Review of Polynomials****Polynomial Graphs****Polynomial Characteristics and Sketching Graphs****Factor and Remainder Theorems****DesCartes’ Rule of Signs****Conjugate Zeros Theorem Problem****Solving Polynomial Inequalities****Polynomial Applications**

## Solving by Factoring

**Exponential Functions**

**Exponential Functions****Parent Graphs of Exponential Functions****Writing Exponential Equations from Points and Graphs****Exponential Regressions****Exponential Function Applications****Exponential Word Problems****Solving Exponentials by Matching Bases**

**Logarithmic Functions**

**Introduction to Logarithms****Special Logarithms****Using Logs (and Exponents) in the Graphing Calculator****Parent Graphs of Logarithmic Functions****Basic Log Properties, including Shortcuts****Expanding and Condensing Logs****Solving Exponential Equations using Logs****Solving Log Equations****Applications of Logs****Transformations, Inverses, Compositions, and Inequalities of Exponents and Log**

**Linear Inequalities**

# Pre-Calculus

**Conics: Circles, Parabolas, Ellipses, and Hyperbolas**

**Tables of Conics****Circles****Applications of Circles****Parabolas****Applications of Parabolas****Ellipses****Applications of Ellipses****Hyperbolas****Applications of Hyperbolas****Identifying the Conic**

**Systems of Non-Linear Equations**

**Introduction to Vectors**

## Parametric Equations

**Introduction to Parametric Equations****Parametric Equations in the Graphing Calculator****Eliminating the Parameter****Simultaneous Solutions****Applications of Parametric Equations****Projectile Motion Applications****Parametric Form of the Equation of a Line in Space**

## Sequences and Series

**Introduction to Sequences and Series****Summary of Formulas for Sequences and Series****Sequences and Series Terms****Explicit Formulas Versus Recursive Formulas****Arithmetic Sequences****Geometric Sequences****Writing Formulas****Arithmetic Series****Summation Notation****Geometric Series****Applications of Sequences and Series**

## Binomial Expansion

# Trigonometry

**Right Triangle Trigonometry**

**Angles and the Unit Circle **

## Linear and Angular Speeds, Area of Sectors, and Length of Arcs

**Graphs of Trig Functions**

**Transformations of Trig Functions**

**T-Charts for the Six Trigonometric Functions****Sine and Cosine Transformations****Cosecant and Secant Transformations****Tangent and Cotangent Transformations****Transformations of all Trig Functions without T-Charts**

**The Inverses of Trigonometric Functions**

**Introduction to Inverse Trig Functions****Graphs of Inverse Functions****Evaluating Inverse Trig Functions – Special Angles****Transformations of the Inverse Trig Functions**

**Solving Trigonometric Equations**

**Solving Trigonometric Equations Using the Unit Circle****Solving Trigonometric Equations – General Solutions****Solving Trigonometric Equations with Multiple Angles****Factoring to Solve Trigonometric Equations****Solving Trigonometric Equations on the Calculator****Solving Trig Systems of Equations****Trigonometric Inequalities**

**Trigonometric Identities**

**Reciprocal and Quotient Identities****Pythagorean Identities****Solving with Reciprocal, Quotient and Pythagorean Identities****Sum and Difference Identities****Solving with Sum and Difference Identities****Double Angle and Half Angle Identities****Solving with Double and Half Angle Identities****Trig Identity Summary and Mixed Identity Proofs**

**Law of Sines and Cosines and Areas of Triangles**

**Review of Right Triangle Trig****Law of Sines****Law of Cosines****Area of Triangles****Applications/Word Problems**

**Polar Coordinates, Equations and Graphs**

**Plotting Points Using Polar Coordinates****Polar-Rectangular Point Conversions****Drawing Polar Graphs****Converting Equations from Polar to Rectangular****Converting Equations from Rectangular to Polar****Polar Graph Points of Intersection**

**Trigonometry and the Complex Plane**

# Calculus (Differential)

**Introduction to Calculus**

**Limits and Continuity**

**Introduction to Limits****Finding Limits Algebraically****Continuity and One Side Limits****Continuity of Functions****Properties of Limits****Limits with Sine and Cosine****Intermediate Value Theorem (IVT)****Infinite Limits****Limits at Infinity**

**Definition of the Derivative**

**Tangent Line****Definition of the Derivative****Equation of a Tangent Line****Definition of Derivative at a Point (Alternative Form of the Derivative)****Derivative Feature on a Graphing Calculator****Determining Differentiability****Derivatives from the Left and the Right**

**Basic Differentiation Rules: Constant, Power, Product, Quotient and Trig Function Rules**

## Equations of the Tangent Line, Tangent Line Approximation, and Rates of Change

**Equation of the Tangent Line****Equation of the Normal Line****Horizontal and Vertical Tangent Lines****Tangent Line Approximation****Rates of Change and Velocity**

**The Chain Rule**

## Implicit Differentiation and Related Rates

## Curve Sketching

**Extreme Value Theorem, Rolle’s Theorem, and Mean Value Theorem****Relative Extrema and First Derivative Test****Concavity and the Second Derivative****Curve Sketching: General Rules**

## Optimization

## Differentials, Linear Approximation and Error Propagation

This was VERY helpful! I had a test just tomorrow in Pre-Algebra and I was practically clueless. I am beginning to understand the areas of math we are going through, though I currently have a D in my class because the 13/20 on my unit test lowered it from a C+. I was about to give up, but I knew acing my next test would help my grade a lot. I recommend this site to any girls that are struggling with math or just need a little re-learning. Wish me good luck on tomorrows test! I hope I get this one, and I will be happier than ever…I’ve always wanted to show my mom a great math test score… 🙂

Thanks for the sweet comment, Emme! I’m thinking good math thoughts your way, and I know you can do it!! Thanks for visiting the site and let me know if you ever have any math questions!! Lisa 🙂

Really, it is very helpful to explain and understand your way. Keep on 🙂 waiting for the new materials.

I just wanted to say thank you! I homeschool my autistic daughter and this site has been very helpful.

Renee,

Congratulations for home-schooling your precious daughter. Thanks so much for kind words and let me know if there’s anything I can do to make the site better 🙂

Lisa

This site is exactly what I’ve been looking for to help my mathematically talented but easily bored son. The examples and explanations are thorough enough to educate but concise enough to hold interest.

Diane,

Thank you so much – you just made my day! Please let me know if there’s anything I can add to make it better 🙂

Lisa

This web-site is very helpful for problems…

One thing to do for make site perfact is : make comment editable…

I just added a plugin for editing comments – thanks for the suggestion!! Lisa

I’m having a really hard time when it comes to radicals 🙁 Too bad, I can’t find something here that will help me. 🙁 maybe you should add them to your site. 🙂 Thank you! Your website is so cool

Hi!

You might look

hereorhereorhere.Sorry you can’t find them here – can you give me an example of a problem and I can see if I address that? I probably just don’t know label it very well. Thanks! Lisa

I also have a hard time with that we just started radicals and it’s confusing

I’m working on “Laplace Transform”…

I need some real life examples which can be solve by L.T.

can you help me ?

please make it faster.

Sorry, I can’t help you with Laplace Transforms yet 🙂 Lisa

dz link z 1daful…….nyc1 dere

i av a question dat i dont really get,can u help me with it

Sure – send it to me! Lisa

I love the site and how much has gotten done since I last looked, but I also see that there isn’t a geometry section. Could you work on this after your other projects?

Hi Tyler! Thanks for writing!! YES – I hope to do a Geometry section, but it won’t be for awhile ;( Hope all is well! Lisa

A cubic polynomial function f(x) has leading coefficient -2 and intercepts the y-axis at 2.If f(1)=1, and f(-2) =-2, find f(-1) and write the complete

function.

You can solve this using systems. let y = -2x^3 + ax^2 + bx +c. Plug in (0, 2) to see that c = 2. Then we have y = -2x^3 + ax^2 + bx +2. Plug in (1,1) to see that 1 = a + b. Then plug in (-2,-2) to see that -10 = 2a – b. Solve that system to see that a = -3, and b = 4. So we have y = -2x^3 – 3x^2 + 4x +2. So then f(-1) = -3. Hope that helps! Lisa

thank you Lisa, you are so helpful.

I gave this link to my Elementary Functions/Trigonometry class. I hope they use it 😉

Thanks so much for writing, and please spread the word about She Loves Math. This makes me want to keep writing 😉 Thanks again, and let me know if I can make it better, Lisa

Lisa

I stuck on trying to figure out how to parametrize

y=(-x^2/72)+x so as to get x=24 sqrt[2] t. Once I have the x

I can figure the y which is -16t^2 +24Sqrt[2] t.

(sqrt is abbreviation for square root)

Thank you for your help

Manuel

Thanks for writing, Manuel. I think when you go from an equation with x and y to parametric equations with t, t could basically be anything. But if they wanted the y in the form y = -16t^2 + something, we could set -x^2/72 to -16t^2 (the first term of the y), and solve for x to get 24sqrt(2)t. Does that make sense? Lisa

Thank you very much. I’d like to work this out a bit to be sure I understand and then, if I may, get back to you.

Needless to say you are terrific.

Manuel

Lisa

As you said, I could have used any parameter for x.

I was just wondering where the author chose

24Sqrt[2] t, from the rectangular equation y=x-x^2/72 to get

y=-16t^2 +24Sqrt[2] t. I tried all kinds of parameters for

y=x^2/72 and sure enough, while I get different values for y,

Thanks again. Basically, I didn’t understand what was meant by

parametrization.

Manuel

Thanks for the explanation, Manuel! Let me know if I can help again! lisa

Hello Lisa

Please excuse if you received this comment before.

I was wondering why you haven’t recommended the use of I believe the best calculator available, namely, Wolfram Alpha. It can be downloaded thru your app in your iPad or from the internet for free. It’s better on the iPad. I think it would help to take the drudgery out matrix`manipulations, vectors, quadratic equations, 3D graphing, et al.

As far as I’m concerned you are the best Just thought you may be interested.

You probably know this anyway.

Manuel

Manuel,

Thanks for writing! I don’t have a lot of experience with Wolfram Alpha and just downloaded the app on my ipad. It looks fantastic! I have used Mathematica in the past; I guess it was based on this.

Thanks so much for the suggestion; that’s a great idea! They don’t use it at the high school, so I haven’t had to help any kids on it, but it’s definitely on my to-do list.

Thanks again and spread the word about She Loves Math 🙂

Lisa

I am having trouble with this section, if you could get back to me asap it would be greatly appreciated.

1)In the nineteenth century, the Austrian monk Gregor Mendel noticed while crossbreeding plants (peas in particular) that often a characteristic of the plants would disappear in the first-generation offspring but reappear in the second generation. He theorized that the first-generation plants

contained a hidden factor (which we now call a gene) that was somehow transmitted to the

second generation to enable the characteristic to reappear.

As an example, suppose we denote the gene that produces the yellow seed by Y and the gene that

produces the green seed by g. The uppercase Y indicates that yellow is the dominant gene and

the lowercase g indicates that green is recessive. The table below shows the possible theoretical

outcomes that can occur when we cross two first-generation plants.

First Generation Plant

Y g

First Generation

Plant

Y YY Yg

g gY gg

Notice from the table that of the 4 possible outcomes, 3 of the plants will be yellow (since 3 have

the dominant Y gene) while 1 of the plants (since only 1 has two recessive genes) will be green.

From this we can say that the theoretical probability of the second-generation plant being green

is ¼ or 0.25.

There is of course a difference between theoretical and experimental. The following table lists

some of the actual results that Mendel obtained in his experiments in crossbreeding peas.

Experimental Results

Characteristics That Were

Crossbred First-Generation Plants Second-Generation Plants

Tall versus Short All tall 787 tall

277 short

Smooth versus Wrinkled

Seeds All smooth seeds 5,474 smooth

1,850 wrinkled

Yellow versus Green Seeds All yellow seeds 6,022 yellow

2,001 green

Notice from the results that tall, smooth, and yellow are all dominant genes. Also notice that based on the experimental results, the probability of a second-generation plant being green is 0.2494. This agrees fairly well with the theoretical probability.

From the tables of possible theoretical outcomes and experimental results, answer the following questions.

1) Assume that we are crossbreeding genetically pure tall plants with genetically pure short plants. Create a table (like the one for yellow and green plants) indicating the possible combinations of tall and short peas.

2) What is the theoretical probability that a plant will be short?

3) What is the experimental probability that a plant will be short?

4) How do theoretical and experimental probabilities compare?

5) Assume that we are crossbreeding genetically pure smooth-seed plants with genetically pure wrinkled-seed plants. Create a table (like the one for yellow and green plants) indicating the possible combinations of smooth-seed plants and wrinkle-seed plants.

6) What is the theoretical probability that a plant will have smooth seeds?

7) What is the experimental probability that a plant will have smooth seeds?

8) How do theoretical and experimental probabilities compare?

Thanks for writing! I’m sorry – I’m having trouble making sense of the tables; let me see if I can find this problem online somewhere. Sorry about that. Lisa

I THINK I understand the problem statement and questions. I have re-stated the problem and answered the questions in a file on my website. I hope it helps. The document is available on http://www.PaulkUSA.com. Click on the “Math Docs” tab and look for the Algebra 2 file named “Statistics Example 01.”

Solution was updated 4/15/15 at 7:00p

thanks a lot for your hardwork

Hi dear just wanna say thank you for the tutorial they are so useful and as a girl I love Maths too!

How long will the calculus site in construction?? coz i hope it will help me in my calculus project assignment. i hope it will settle faster, pleaseeeee

Thanks for writing! I’m SO sorry about not completing the Calculus portion yet – I hope to by the fall! Keep checking back! Lisa

A great resource, you cover almost everything and I see how much work you have put in. Thank you very much and I will be coming back here again

Thanks so much for the nice comment; I’ll work harder to finish the site 😉 Lisa

this problem makes me confused, can you help solving this?

a receptacle contains x ounces of acid. A second receptacle contains x ounces of water. From the acid y ounces are removed, placed in the water, and the contents thoroughly mixed.Then, from the second receptacle, y ounces of the mixture are removed and placed in the acid. find (a)the concentration of of acid in the second receptacle, and (b)the concentration of water in the first receptacle

Here’s how I’d do this one – I got y/(x+y) or y/(x+y)*100 % for concentration of acid in the 2nd receptacle. Then I get (y * (x/(x+y))/(y+x-y) = y/(x+y), or y/(x+y)*100 % for the concentration of water in the first receptacle (amount of water over total amount). You can try it with real numbers. Does that make sense? Lisa

I’m having trouble at this problem… hope you reply..

Two pipes running simultaneously can fill a swimming pool in 6 hours. If both pipes run for 3 hours and the first is then shut off, it requires 4 hours more for the second to fill the pool. How long does it take each pipe running separately to fill the pool

That’s a great problem! I actually found it solved here: http://www.softmath.com/algebra-word-problems/show.php?id=18373

Please add log inequalities (for pre cal)

there’s a problem that I can’t seem to figure out:

log (12x) + log (X-1) > 2

would the domain restriction be X>1 or would it be the domain restriction of the two log combined – log (12x^2 – 12x?

THanks for writing Victoria; I’m just about to put these types of problems on my web page. For the domain restriction, you have to check each log argument, so we have 12x>0, so x>0 and x-1>0 so x>1. Therefore domain restriction would be x>1. For the whole answer I get x>(3+sqrt(309))/6.

Here is a video solution:

https://m.youtube.com/watch?v=2R0E-t_AQ74&feature=youtu.be

Here is another video solution (graphic)

https://m.youtube.com/watch?feature=youtu.be&v=UZBj_B5Kk4E

Math article by Nghi Nguyen:

URL

http://www.shelovesmath.com/wp-content/uploads/2016/02/The-Transposing-Method-Shortcut-in-solving-algebraic-equations.pdf

Hello Lisa,

I love your website. It would really be helpful if you could cover non-differential Calculus for us as well!

Thanks so much for your kind words – comments like these make me want to keep writing! I will get to non-differential Calculus eventually – hopefully this summer! Please spread the word about shelovesmath.com 🙂 Lisa

This website is amazing! Even though I am a boy (and 42 years old), I just want to say I really appreciate all the hard work you have put into it. I wanted to brush up my skills a bit for a standardized test I have to take soon for nursing school, and this is by far the most comprehensive, complete, logical review I’ve seen. Exactly what I needed! Thank you.

David,

Thanks so much for writing!! And kudos to you for being a nurse – such a wonderful field. I’ve thought about making the site more generic – not “she loves math” but something else – so still may. Let me know if you have any more feedback on the site! Lisa

Hello Lisa,

I looked through your website and I couldn’t really find anything that dealt with Binomial Series. I am having trouble with using the alternate definition of binomial coefficients to show that if n is a non-negative integer then (n k) = 0 whenever k > n. How does this allow me to express the binomial expansion as the infinite series (a+b)^n = (summation from k=0 to infinity) Σ (n k) (a^n-k)b^k

THanks for writing! Did you see this section? http://www.shelovesmath.com/precal/binomial-expansion/ I guess it doesn’t address the series going to infinity though – I need to look into that. Thanks, Lisa

Excellent resources.

Well done.