Linear Functions and Graphs

The content of this article may be too opinionated according to the guidelines. This article needs to be rewritten to be more formal and a little less tutorial.

In this section, some topics from Algebra 1 will be reviewed. These topics include graphing linear equations and inequalities and observing linear models.

The x-y Plane

The x-y plane is a two dimensional coordinate plane. A plane is like a universe in two spacial dimensions rather than our universe of three spacial dimensions. To designate location in this plane, two axes are used. They are the x-axis and the y-axis both represented by lines. The x-axis runs horizontally and the y-axis runs vertically. These to perpendicular lines intercept at the Origin. Number markings are evenly spaced on these axes. Using the numbers, coordinates can be identified. A coordinate is the x location and y location and are represented by (x,y). Observe the graph below.

coordinate.png

This image is a coordinate plane. Notice the x and y axes. The plane is divided into four Quadrants by the axes which are characterized by the sign of the units of its (x,y) pairs. Simply, Quadrant I will only contain points with (+,+) pairs, Quadrant II will have (-,+) points, Quadrant III will contain (-,-) points, and Quadrant IV has (+,-) points. The origin O is where the axes intercept, or at coordinate (0,0). Every mark on this graph is one unit apart; some coordinate graphs have dilated axes which means that the marks are either less than or greater than one unit apart. Let's observe the points on the graph. Point A is located at coordinate point (6,3). This means that it is 6 units in the positive horizontal direction (right) and 3 units in the positive vertical direction (up). Point A is in Quadrant I since it has a (+,+) pair. Point B is located at (-8,7). To get to that point, you have to go 8 units left and 7 units up. Point C is located at (-3,-5). Where is point D located?
A coordinate graph can help graph functions which is an equation using a variable to define the other.

Linear Functions

A linear function is a first degree equation of two variables x and y where x is the independent variable and y is the dependent variable. In other, simpler words, a linear function is an equation that forms a line on a coordinate plane. y can also be written as the f(x). A linear function describes a rate for x. For example, in the equation y=2x, 2 is the rate by which y increases or decreases depending on the value of x. This is the slope of the line generated by the equation. Let's discuss the components of a line in detail.

Slope

The slope of a line (its rate) is defined by its rise over run. Mathematically, the slope is the change in y over the change in x. The slope is the amount that the y changes as x changes, simply. The slope of a line can be identified using any two points. Using the points' x and y components, the slope can be identified using a simple equation.

(1)
\begin{align} m = {y_{1}-y_{2} \over x_{1}-x_{2}} \end{align}

In this equation, m is the slope, and the x's and y's are the points' components (x,y). In the real world, slope is found everywhere from interest rates to gas per gallon. Anything with a constant rate has a mathematical slope.

A linear function can have three different types of slope. A positive slope is characterizes by a line emerging from the lower left part of the graph and exiting off the upper right. Positive slopes happen with increasing rate. A negative slope is the opposite of a positive slope where it enters from the top left corner and exits the bottom right. In the case of a negative slope, the rate of change is decreasing. Zero slope is when the line is parallel with the x-axis, or completely horizontal. In this case, there is no rate of change meaning that no matter what the value of x is, y is the same. There is a fourth slope, but it cannot be formed by a linear function. A vertical line parallel to the y-axis has an undefined slope since division by zero must take place.

Intercepts

The intercepts of a linear function are where the line crosses the x and y axes. Any line can only cross each axes once. Because of this, there is one x-intercept and one y-intercept. The intercepts of a line help characterize a linear function, and in the real world an intercept can represent where payments are over or minimum cost for something.

The x-intercept is the point on the line where y is equal to 0. With any function, to find the x-intercept, substitute y with 0 and solve for x. The coordinate pair for the x-intercept is (x,0).
The y-intercept is where x equals 0. To find the y-intercept of a linear function, plug in 0 for x and solve for y. The coordinate pair for the y-intercept is (0,y).
Note: The intercepts of a linear function (or any function) are officially coordinate pairs, and teachers will expect them this way!

Graphical Example

Let's observe the following graph:

coordinate2.png

On this graph, there are two lines. Line 1 has a positive slope and Line 2 has a negative slope. Let's identify the intercepts of Line 1. What is the x-intercept of Line 1? The x-intercept is where y equals 0, or where the line crosses the x-axis. The line crosses at the -1 mark on the x-axis. Putting it as a coordinate pair, the x-intercept of Line 1 is (-1,0). So, what is the y-intercept? The y-intercept is where x equals 0, or where the line crosses the y-axis. The line crosses the y-axis at the 2 mark, meaning Line 1 has a y-intercept of (0,2). Now, let's find the slope of Line 1. To find the slope, we can use two methods: rise over run or our equation for slope. For Line 1, we will use our equation using the two intercepts as our points. Remember the slope equation? (1) Plug in the points into the equation. We get ${2-0 \over 0-(-1)}$ which simplifies to a slope of positive 2. For Line 1, the slope is 2, the y-intercept is (0,2), and the x-intercept is (-1,0). Now, let's observe Line 2. What are the intercepts of the line? Find where the line crosses the axes. It crosses the x-axis at 4 and the y-axis at 3. So, Line 2's x-intercept is (4,0) and y-intercept is (0,3). Now let's find the slope. In our previous example, we found the slope using a special equation and the two intercepts. This time, we will find the slope using rise over run. To use this method, find two easy to identify points. These can be the intercepts, but for this example, we will use the x-intercept, (4,0), and another point on the line, (8,-3). Find out how many units up or down you must go to reach the point. In this case, it takes three units down, or -3 units. Now, find how many units you must go to reach the point laterally. For Line 2, you must go 4 units to the right. Take your first number over your second number for a slope of -3/4. So, Line 2 has a slope of -3/4, an x-intercept of (4,0), and a y-intercept of (0,3).

The Three Forms

There are three forms for writing a linear function as an equation. They are the slope-intercept form, point-slope form, and standard form. Each equation uses the variables x and y which represent the location on the axes of a graph. A line is typically expressed using an equation in one of these forms. Why an equation? The equation tells us the properties of the line without us having even to look at it. Using just the equation of a graph, the intercepts and slope can be identified. Now, we will describe each form in detail including the way how to use the equation to graph a line and using a line to produce an equation.

Slope-Intercept Form

This is by far my favorite form. This equation is solved for y meaning the value of y can be calculated arithmetically with any value of x. This form directly exposes the slope and y-intercept which is all you need in order to graph an equation. Here is the equation.

(2)
\begin{equation} y = mx+b \end{equation}

In the equation, m is the slope and b is the y-intercept. The x and y are the input/output variables. One note about the b variable: Don't forget the sign! If there is an equation like y=2x-3, the y-intercept is (0,-3).

So, how do we turn this equation into a graphed line? Using are equation, y=2x-3, we should first consider the y-intercept. Immediately, the y-intercept is known, so we can draw one point immediately. Using the slope, 2, we can find another point. However, when dealing with slope, it is best to put it in fraction form. So, our slope is actually 2/1. For any slope, translate up the amount of the numerator (if negative, you must go down) and translate right the amount of the denominator (if negative, go left). So, using this general application, we can draw a new point at (1,-1). With two points, a line can be drawn.

Now, let's say you need to convert a graph into an equation. First, observe the graph. Find the graph's y-intercept which gives you the value of b. If the y-intercept is not obvious, then select any two points on the graph that are. Remember, with any two points, you can form the equation for a line. One you have the y-intercept and another point or just two points, use your slope formula to find the slope m. (1) When you have the slope, plug it into the slope-intercept form. (2) If you already identified the y-intercept, then plug the y part into b and you're done. If not, plug the x and y components of one of the two points into the form along with the slope and solve algebraically for b.
Graphing calculators use this form for graphing.

Point-Slope Form

This form is very useful. Let's say that a problem gives you the slope and one point of the linear function. The point-slope form is meant for this purpose. The form incorporates the slope and room for a point which can easily be algebraically solved into Slope-intercept form. Here is the form:

(3)
\begin{equation} y-y_1 = m(x-x_1) \end{equation}

This time, a new x and y are introduced with the subscript 1. These represent the point components. This equation shows the slope m and the point (x1,y1). When the variables have been filled in, you can algebraically solve for y to put the equation in slope-intercept form or solve for the standard form (which comes next).

It is simple to turn an equation in point-slope form into a line on a graph. Immediately, you know one of the points, and with the slope, you can find a new one like in the previous form. Take the linear equation y-3=2(x+2) for instance. First, identify the point the equation displays. Find x1 and y1. In this equation, the point is (-2,3). Beware the signs! Since the form uses subtraction, the point numbers in the equation are the opposite of what they actually are. Now find the slope m. Use the slope to find a second point and draw the line.

Now, let's try to put a line into this form. Find any two points that are on the line. These points could be any point. Using the points, find the slope with the slope equation. (1) Now, use any of the two points to fill in x1 and y1 in the equation. The slope fits under m. After that, you're done. The point-slope form is very useful in problems that give you the slope of a line and a point.

Standard Form

The final linear function form is the standard form. This form is seen very often because it is easy to read and, well, standard. Standard form cannot have fractions in it. So, the general description of a Standard linear equation is the sum of the integer amounts of x's and y's to equal an integer constant. Here's the form.

(4)
\begin{equation} Ax+By=C \end{equation}

Remember, the values A, B, and C can only be integers! This form is seen a lot, and knowing this form is an essential for Algebra. The values A, B, and C are integers probably for the reason of neatness. The variables x and y are in the equation as always, except they are together this time. In standard form, the slope and intercepts are not obvious, and if you want to identify slope, putting the equation into slope intercept form.

Even though the slope or intercepts are not obvious in standard form, an equation in this form is still able to be graphed easily. A graph only needs two points to be graphed, and the intercepts of an equation can be easily found. As you should know, to find the x-intercept, substitute 0 for y, and to find the y-intercept, substitute 0 for x. Let's take the equation 2x+3y=6. Find the x-intercept: substitute 0 for y. 2x=6 After simple solving, x=3. The x-intercept is (3,0). Find the y-intercept: substitute 0 for x. 3y=6 After more simple solving, y=2. The y-intercept is (0,2). With these two points, a line can be drawn.

To put a graph of a line into standard form, it is best to put the line into slope intercept form first. Find the y-intercept and the slope. Now, you can algebraically solve for the standard form. Subtract mx on each side. At this time, remove all fractions and decimals by multiplying the equation by the appropriate numbers, presumably the denominator of the fractions. For example, 2/3x+y=1/2 is not in standard form because not all the coefficients are integers. Multiply each side by the denominators of the fractions. In this example, you would multiply by 6 (2*3) to yield an answer of 4x+6y=3.

Graphical Example

coordinate3.png

Here is the graph of a linear function. Create an equation in each form for this line. Ok, let's begin.
First, we will put the line into slope-intercept form (2). First, can we tell what the y-intercept is? In this line, the intersection is obvious. The y-intercept is (0,2). The y-intercept fits into the value of b. Now, to find the slope, find another point. Observe the point in Quadrant II. What is the coordinates for that point? The coordinates are (-7,4). Use the slope formula (1) to find m. The slope, after plugging in what we know and using simple arithmetic, is -2/7. Now, since we have m and b, we can put these values into slope intercept form. The equation is y=-2/7x+2. Now, let's find the point-slope form (3). We already have the slope m, -2/7. Use a point on the graph to identify x1 and y1. Using the point (-7,4), we get y-4=-2/7(x+7). Finally, we'll find the standard form (4). Using our slope-intercept form, we can algebraically solve for the standard form. Subtract the x term first. This yields 2/7x+y=2. Note that we are still not done: there is a fraction. To get rid of the fraction, multiply by the denominator on both sides. Multiplying by the denominator causes the fraction to cancel. After that, we get our final linear equation 2x+7y=14. Our three equations are:
y=-2/7x+2
y-4=-2/7(x+7)
2x+7y=14

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