# Algebra

### What is Algebra?

Algebra is a broad part of mathematics that deals with unkown quantities called variables. Some of the topics studied in College Algebra are:

• Polynomials
• Factoring
• Equations
• Systems of equations
• Sequences
• Graphing

The UNT Dallas Learning Commons can help students expand their mathematical knowledge and grasp the concepts of algebra. Don't hesitate to come in for help!

### Functions

In algebra, we can think of a function as a machine that takes numbers in, and outputs numbers. The number that is spit out by our function/machine is related to the number that is inputted. We write most functions in the form of:

$$f(x)="\dots"$$

$f(x)$ means that the function $f()$ is dependent on the input variable $x$. In other words, when we put $x$ into $f()$ we get $"\dots"$ out.
It is important to note that functions are also written using $y$ instead of $f(x)$, but they both represent a function dependent on another variable. Lets try out some more examples.

#### Example

Take the function $$f(x)=x^2$$ Remember that $x$ is a variable that can represent any number.
So if we plug in $2$ for $x$ , we get $$f(2)=2^2\\f(2)=4$$ Not all functions are as simple as the one we just explored. You might encounter some functions that look like $y=12x^3+2x^2+6x+10$ or $y=e^{x^2}$, but these all are functions that take an input and output a number.

### Slope

Slope is defined as the 'steepness' of a line. Another common definition of slope is the rise over the run of a line/function. Mathematically, slope is the change in the $y$ direction over the change in the $x$ direction.

$$\text{Slope}=\frac{\text{rise}}{\text{run}}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}=m$$

So why is slope important? Slope is important because it is related to the equations of lines.

#### Example

Find the slope of the line going through the points $(\overset{x_1}{1},\overset{y_1}{5})$ and $(\overset{x_2}{-2},\overset{y_2}{-1})$

Using the equation for slope $$m=\frac{y_2-y_1}{x_2-x_1}$$ we can plug in our points to obtain $$m=\frac{{-1}-5}{{-2}-1}$$

After some subtractions, we get: $$m=\frac{{-6}}{{-3}}$$

Simplifying gives us:$$m=2$$

### Equations of Lines

A line is defined as a one dimensional object with no width and infinite length. It has no bends or wrinkles and it is the shortest path between two points. A line is can be expressed as an equation. The most used form of the equation of a line is called the slope intercept form:

$$y=mx+b$$

where $m$ is the slope of the line and $b$ is the $y$-intercept.

#### Example

Take: $$y=4x+5$$ From this equation, we can tell a few things about the line it represents.

• The slope$(m)$ of the line is $4$
• The $y$-intercept is $5$

Another way of describing a line would be using it's point-slope form: $$y-y_1=m(x-x_1)$$ In which $x_1$ and $y_1$ are a set of coordinate points on the line and $m$ is the slope of the line.

#### Example

Write the point slope equation of a line going through the points: $(1,5)$ and $(-2,-1)$

First we need the slope of the line. From the first example, we know the slope of this line is $2$.

Now, we just need to plug in our slope and one of our points into our point slope form equation. For this example we will use the point $(1,5)$ but the point $({-2},{-1})$ will also work.

We now have: $$y-5=2(x-1)$$

From here, getting to the slope intercept form is easy. Applying the distributive property gives us: $$y-5=2x-2$$

Rearranging terms leads us to: $$y=2x+3$$ And now, we are in slope intercept form.