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.