Making Connections Between Algebraic and Graphical Representations

General form equations (ax + by + c = 0), standard form equations (ax + by = c), and slope-intercept form equations (y = mx + b), all create a straight line, a line that goes directly between two points without any deviation or holes and continues onward in both directions forever.

The slope-intercept form is the easiest method to use for graphing as it provides the slope and the y-intercept.

Accepted Foundational Truths

Two points determine a line (points and line have no thickness, but mark and show connectedness).

The shortest distance between the two points is called a line if continuation in both directions occurs after passing through the two points without end. Continuation forever is shown with an arrow at both ends. On graphs this may also be shown by drawing the line to the edge of the graph in both directions. The domain and range of a line is all real numbers.

A line segment (segment for short) occurs if no continuation occurs past the points. A segment has a domain and range that is constrained, restricted between the two points.

A ray occurs if there is continuation in only one direction. The ray has a restricted domain at one point and is continuous forever through the second point.

Image examples of a line, segment, and ray

Essential Background Knowledge

Where are points located on a coordinate plane?

Points are found on a coordinate plane and are properly shown using an ordered pair of numbers with the horizontal movement from the origin (0, 0) first, followed by vertical movement second. The ordered pair looks like (x, y) where x is the horizontal movement and y is the vertical movement.

The coordinate is any ordered pair (a, b) that provides the location of the point on the graph. Sometimes the ordered pair is called a coordinate pair because the point lies on the coordinate graph, a graph with grid lines representing the units of measure on the graph.

Let's examine the graph of the line that runs through these two points. Note that the line goes through point A (3, 5) and point B (−4, −2).

To locate point A (3, 5). Move right 3 horizontally, then up 5 vertically.

Point B (−4, −2) is plotted by moving horizontally left 4 (the negative sign indicates left) and down 2 (the negative sign on this number means down).

graph example: The y-intercept of the line is 2, the point (0, 2)

The y-intercept of the line is 2, the point (0, 2). The y-intercept is where the line crosses the y-axis. The line intercepts the y-axis at 2.

The x-intercept of the line is -2, the point (-2, 0). The x-intercept is where the line crosses the x-axis. The line intercepts the x-axis at -2.

What is Slope?

Refer to the graph above. Note that from any whole number point to the very next, you can move right one unit and up one unit and reach the next point that is defined on the whole number grid lines. Check this out: Locate point B (−4, −2). Move right one unit and up one unit. Do you see that you are now at another point (−3, −1) and if you repeat the pattern again you are at the point (−2, 0)? If you continue the pattern you will eventually reach point A (3, 5). This repeated pattern on a straight line of finding the next cross gridline point is called the slope of the line.

Slope is incline between two points given. The slope is found by dividing the change in the y-values by the change in the x-values. The slope formula is: m = where m stands for slope.

Example

How is slope found without a graph from the two points given above, point A (3, 5) and point B (−4, −2)? The slope m of the line

y = x + 2 can be determined by using the slope formula.

m = =

The slope of the line is 1 unit, meaning = which means the line has a positive rate of increase of 1 unit up and 1 unit to the right.

image of rise over run. y's rise on top and x's run flat

How do I remember the slope formula meaning?

Check your knowledge of slope by answering each of the following questions.

Hint:

Given the two points (x_{1 ,} y₁) and are you remembering to use the y₁ and x₁ vertically over each other in the slope formula? Also, y₂ and x₂ should be vertically aligned as well. Last check, did you subtract? Find the slope of each line.

What does a positive slope mean and can you draw an example?
What does a negative slope mean and can you draw an example?
What does a zero slope mean and can you draw an example?
What does an undefined slope mean and can you draw an example?

Several of the words that we just learned are essential vocabulary words

increasing (positive) slope - an increasing rate of change, rising from left to right
decreasing (negative) slope - a decreasing rate of change, falling from left to right
undefined slope - no rate of change, vertical with no horizontal movement
zero slope - zero rate of change, horizontal with no vertical movement

Algebraic and Graph Analysis

Equations and graphs explain information that is important. In the following video and Graph Analysis handout, information will be determined graphically. Linking the graphical representations to the algebraic equations will allow you to determine the information regardless of how the information is presented.

Carefully examine the next four graphs to identify the various line information that can be determined from a graph. Note the equations for a horizontal line and a vertical line.

The horizontal line equation is always y = a number, the number is the y-intercept.

The vertical line equation is always x = a number, the number is the x-intercept.

How to Remember?

A quick way to remember which is which: Call the line by the axis the line runs into. The x = lines run into the x-axis and the y = lines run into the y-axis.

Graph example of y = x + 3
y - intercept = 3
y - intercept point (0,3)
Slope m = 1
positive slope implies increasing graph
positive rate of change
x - intercept = -3
x - intercept point (-3, 0)
up 1, right 1 Graph example of x = -x + 4
y - intercept = 4
y - intercept point (0,4)
Slope m = -1
negative slope implies decreasing graph
negative rate of change
x - intercept = 4
x - intercept point (4, 0)
down 1, right 1 Graph example of y = -2
y - intercept = -2
y - intercept point (0, -2)
Slope m = 0
zerp slope implies horizontal line
no rate of change
x - intercept = none
x - intercept point none
flat, no rise or change in y values

Please NOTE the necessary corrections for the x=5 graph below:

One of the characteristics is that it has a negative rate of change. This is incorrect. It doesn't have a rate of change.
It says the x-intercept is 4 and the x-intercept point is (4, 0). This should be 5 and (5, 0).

graph of:
y intercept: none
y intercept point: none
Slope m=undefined= y1-y2 over 5-5=# over 0
undefined slope implies vertical line
negative rate of change
x-intercept: 4
x-intercept point: (4,0)
describe the slope from one grid point to the next vertical, no run of change in x-values

Review analysis for a line graph or slope-intercept form equation

The slope-intercept form of the line is given by the equation, y = mx + b.

The m stands for the slope of the line.

m is positive, the line is increasing from left to right
m is negative, the line is decreasing from left to right

The x and y and are real numbers, an ordered pair (x,y).

The y of the ordered pair is determined by inputting an x value into the equation with m and b values. Evaluating the expression mx + b provides the value of y and the associated ordered pair (x,y). Using the equation y = x + 2 from Illustration 2, if x is −1 then y = - 2 + 1 and solving the equation y = 1. Therefore the point on the line when x = -1 is (-1,1).

The letter b stands for the y-intercept, the y-value where the line crosses the y-axis (vertical axis).

Suppose the y value is known and not the x value of the equation. Can the y value be found? Yes, simply place in the equation the y value with the m and b values and solve for the x value. Suppose y = 6.

6 = x + 2

Put 6 in the equation for y

4 = x

Subtract 2 from both sides

Graphically without an equation all of the above may be determined by examining the points and slope, particularly those points that cross gridlines on the graph. Graphs without easily determined gridline points will require a calculator to use with the equation.

Inequality Graphs Analysis

The difference between a linear equation graph, y = mx + b, and the linear inequality graph, y ≤ mx + b, for example, is the change to the inequality sign. The inequality signs that may be used are: ≤,≥, <, >, ≠. Each of these signs implies that there is more solutions than those on the line formed from the linear equation.

A linear equation graph is just a line, nothing more. The points on the line are the graph.

An inequality graph may or may not include the line, but it includes an infinite amount of points. If the linear inequality does not include the line, then only the symbols <, >, ≠ would be used.

The inequality graph shown in the illustration below indicates the red downward arrows that the solution is to include everything below the line.

graph of y is less than or equal to x plus 2

Thus the solution includes the line and all points under the line. Use the y-intercept to determine whether the shading is above or below the line. If the shading is above the line, then use ≥ (> if the line is dashed) and if the shading is below the line, use ≤ (< if the line is dashed).

Use the Graph Analysis Inequality Video and the Graph Analysis - Inequality handout to examine various equations and graphical representations of inequality graphs.

Examine the number line below. This is an illustration of one-variable inequalities on a number line. Remember that the open circle means the number is not included in the solution.

image of of one-variable inequalities on a number line.

Finding Equations of a Line

Using the Y-Intercept Form for the Equation of a Line

y = mx + b

Example: Given points C (8, -3) and D (-4, 5) find the equation of a line.

Step 1. Find the slope between the two points.

m = = = = -

On a graph, this slope means that the line is decreasing from left to right (the negative sign), falling 2 and moving to the right 3.

Step 2. Use the slope-intercept formula with the slope and one of the points and solve for the y-intercept, b.

y = mx + b

5 = - (-4) + b Substitute the slope and the point into the equation.

15 = -2(-4) + 3b Multiply each term by 3 to clear the denominator.

15 =8 + 3b Simplify.

7 = 3b Combine like terms by subtracting 8 from both sides.

= b Divide by 3 to solve for b, the y-intercept.

Step 3. Use the slope-intercept formula with the slope, m, and the y-intercept, b, to write the equation of the line.

y = mx + b

y = - x +

Using the Point-Slope Form for the Equation of a Line

Use the equation of slope to find this formula which is useful when a point and slope are provided or can be determined.

m = equation image indicator

Change the format of the equation to only reflect one input point (x₁, y₁).

m (x - x₁) = y - y₁

Multiply both sides by the denominator.

y - y₁ = m(x - x₁)

Reverse sides so the y's are on the left.

Example: Given point A (3, 1) and slope m = 5, determine the equation of a line. Use the point-slope formula with the given filling in the point and slope given.

y - y₁ = m(x - x₁)

y - 1 = 5(x - 3)

equation in point-slope form

y = 5 (x - 3) + 1

y = 5x - 15 + 1

y = 5x - 14

equation in slope-intercept form

Self Check Problems

Now is the time to check out what we understand concerning slope and finding the equations of lines. This is a big topic that will continue throughout your study of math. Try the following problems, check your answers and the solutions as needed. Answers are provided and solutions for reference as needed.

Finding Distance

Distance Between Two Points

Two points also allow us to find the distance between two points and since we can find the distance, the midpoint between the two points.

To find the distance between two points, use the distance formula

d =

Example: Given point A (4, 7) and point B ( -2, 3) find the distance between the points.

d =

=

=

=

Leaving the answer in square root form is appropriate if the square root is not a whole number.

Midpoint Between Two Points

In the above example, the half distance would be dividing the answer by 2, or square root of 52 divided by 2 . This is not the midpoint. Midpoint means that the (x, y)coordinates are needed. To find the midpoint use the following formula to find it.

midpoint = ,

Example: Given points A (-2, 5) and B (4, 3) find the midpoint.

midpoint = ( , ) = ( , ) = (1 ,4)

The point (1, 4) is the midpoint, the point in the middle between the two given points.

Watch the video to find an endpoint given a midpoint and one endpoint.

The intersection of Graphs for Finding Systems of Equations Solutions

The intersection solution of a system of equations may also be found using a graphing utility. Here are two picture examples.

The intersection of these two lines appears to intersect on a gridline, sort of. Does it? If you put in the point (3, 1) do both equations work?

image of graphing utility y=negativexplus4

Verify:

y = −x + 4

1 = −3 + 4 = 1 (okay)

3y = 2x − 1

or 3(1) = 2(3) − 1 = 6 - 1 = 5

3 = 6 − 1

3 ≠ 5 (Oh no, the point just appears good)

What did one equation working and not the other mean? The point was on the first line, but was not on the second line (not a solution for the second line).

Find the real intersection by solving the simultaneous equations.

Solve

y = - x + 4 -3y = 3x - 12 Multiply by negative 3

3y = 2x - 1 3y = 2x - 1

0 = 5x - 13 Add the equations together

13 = 5x Solve for x

13/5 = x

So, now substitute 13/5 for x in one of the equations (pick the one that looks the easiest) and solve for y.

y = −x + 4

y = −13/5 + 4 = -13/5 + 20/5 = 7/5 = 1 1/5.

Remember, you must read data carefully (yes, graphs have data) to determine whether what you see is true. When in doubt, check!

Here we have a point that is clearer. The point of intersection of the two cars is shown as (2, 1).

image of two cars meeting at a point 2,1

Let's make a quick check with equations that we observe from the graph.

Since the equations are not given, we will find the equations and then verify the point of intersection.

We will want to use the formula y = mx + b for the equation of the line.

Blue line equation:

The y-intercept b = 2, with the equation of the point (0, 2)

m = −1/2 is observed from counting down 1 from the point (0,2), the rise, and across to the right 2, the run, to the next gridline point that the lines intersect (2, 1).

The blue equation then is y = −1/2 * x + 2

or 2y = -x + 4 if we multiply all terms by 2 to clear the fraction.

Checking to verify the intersection point (2, 1)

y = −1/2 * x + 2

1 = (−1/2) * (2) + 2

1 = −1 + 2

1 = 1 √

Green line equation:

The y-intercept b = −1, with the equation of the point (0, −1)

m = 1 is observed from counting up 1 from the point (0, −1), the rise, and across to the right 2, the run, to the next gridline point that the lines intersect (2, 1).

The green equation then is y = 1 * x − 1 or y = x − 1.

Checking to verify the intersection point (2, 1).

y = x − 1

1 = 2 − 1

1 = 1 √

In this case, the intersection point was valid. Note that we determined the equations that made the point valid as well from the graph information.

Notice that the diagram of the lines intersects at a point that is easily discerned if the lines intersect on a grid line. However, if the lines intersect off the grid lines, a graphing utility or a by-hand calculation would be required.

The Green and Blue Car graph above provides a different perspective of graphs and lines. Note that now an analysis of whether the two cars would intersect would be appropriate given the velocity of the cars.

Inequality Graphs

So what happens with inequality equations and graphs. Below is an illustration of inequalities that intersect. The solution set here would be all points that are in both solutions.

In the graph below, notice that the green line is dotted and the blue line is solid. The intersection will include the blue line points but not the green line points as a dashed line indicates points are not included. The intersection is where the green and the blue shading mix.

The intersection can be described as all blue points less than or equal to the line

y ≤ −x + 4 which are also green points that are greater than the line y > 2/3 *x −1. This is the blue-green area of the graph on the left.

image graph of y less than equal to negative x plus 4

To find the intersection of a system of inequalities, the intersection point may be found the same way as equals equations.

3y = 2x − 3 equation image indicator 3y = 2x − 3

y = −x + 4 − 3y = 3x − 12 Multiply by -3

0 = 5x - 15 Add the equations

15 = 5x Solve for x

3 = x

Now input 3 for x to solve for y.

y = −x + 4

y = −3 + 4

y = 1

The intersection point is (3, 1).

To shade the green area since the equation indicates greater than, the green shading is above the point (3, 1), which means every point above the line y > 2/3 *x −1.

To shade the blue area since the equation indicates less than or equal to, the blue shading is below the point (3, 1) which means every point below and equal to the line

y ≤ −x + 4. The intersection solution set of all points that belong to both equations is the points in both sets. All points must be above the line y > 2/3 *x −1 and below or equal to the line y ≤ −x + 4.

Try these questions to see if you understand the inequality graphing.

IMAGES CREATED BY GAVS OR OPENSOURCE