We are going to explore linear equations and how the basic mathematics functions of addition, multiplication, division and composition affect their physical attributes. Let's start with two basic equations and graph them. The first equations is y = x + 4. The second equation is y = 6x - 5. Through out this exploration we will be changing the attributes in the first equation y=x+4, while keeping the second equation y=6x-5 the same. The lines on the graph represent the given equations.
Let's review some of the physical characteristics of a linear equation. The basic equation for a line is y = m+ b. The "m" is the slope of the line with the fraction standing for the rise/run. The rise is the move up the y-axis from the y-intercept. And the run being the move from the y-axis along the x-axis. Now let's apply the knowledge to the given equations. The slope of y=x+4 is 1/1. The slope of y=6x-5 is 6/1. The "b" in the equation is they-intercept. This is the point at which the line crosses the y-axis or the point at which x=0. In the first equation its when y=4. In the second equation it is where y=-5. This will tell us where to put our first point when graphing these lines. The second point can be found by using the slope. In the first equation the slope was 1/1. So starting at point (0,4), you need to go up one point and move to the right one point. This would locate the second point at (1,5) and if we look at the graph of this line it does pass through that point. Now we can check out the second equation. The y-intercept is (0, -5), which can be our starting point. Now the slope was 6/1. So we move up from the y-intercept by 6 and over to the right by one point. This would put our second point at (1,1). This is shown to be true on the graph above. We need to connect the corresponding points and get a graphical display of our linear equations.
Now let's change the first equation to y=x+2 from y=x+4. How do you think this will change the graph? What specific part of the equation was changed?
The y-intercept has been reduced to 2 but the slope has stayed the same. Will these two lines ever intersect? No. They will run parallel to one another with y=x+2 shifted two spaces down the y-axis from the original equation of y=x+4.
What do you think
will happen if we change y=x+4 to y=2x+4?
What was changed? As you probably noticed by now, the slope was
changed from 1/1 to 2/1.
What will happen to the line?
With the increase of the slope the line seems to be "steeper". It moves up the y-axis more quickly. Do these lines intersect? Yes, at the point where x+4=2x+4. Or at the shared y-intercept of (0,4). This is the pivot point for the line when only the slope is changed.
Now that we have seen how a line changes when the slope is changed and the y-intercept is changed, we will now look how the addition of each of these equations to a fixed equation changes the sum. Our fixed equation will be y=6x-5. Then each of the three equations we worked with will be added to it. The original graphs of the equations will be compared to the sum of the two equations and the sum of the original equations. Do you think the sums will shift in the same directions as the individual equations?
Let's do a quick review on adding two linear equations. Like terms are added together. So with our two original equations:
So our new equation is y=7x-1. This still produces a linear equation in the form of y=mx+b. Now we will look at the graphs of the two original lines and the sum of the two.
The graph shows that the slope of the new line is fairly steep. This is because it is going from 6 on one line and 1 on the other line to 7. The y-intercept is changing also by combining a -5 intercept with a +4. This will put the y-intercept at (0,-1).
When we add together y=x+2 and y=6x-5 the sum is y=7x-3. It looks as if because the only change is the y-intercept in the first equation the slope of the sum is the only change. It has shifted down by two and the two summation equations are parallel. This can be seen by looking at the graph and checking random points along the two lines of y=7x-3 and y=7x-1. They will always be the same distance apart, parallel.
Now we should
add together 6x
-5 and 2x + 4. The sum is 8x -1. This will obviously
change the slope and leave the intercept alone.
This will change
the angle to a much steeper slope. What is the new slope looking
at the new equation 8x-1? CORRECT !!! 8/1.
We will not graph it but what would happen if we changed the slope to 1/2 ? the line would become "flatter". What if we changed the y-intercept to +3? The line would intersect the y-axis at (0,3).
We will start back with our basic equations of x+4 and 6x-5, and multiply these together.
Is the new equation
of 6x2+19x-20 still a linear equation?
No. The best indicator of this is the 6x2. . This indicates
the equation intersects the x-axis in 2 places. Let's see what
it looks like.
The product of
forms a parabola with the sides of the parabola crossing the x-axis
at the points where x+4 and 6x-5
cross the x-axis.
We will now see how changing the y-intercept and multiplying it to our base equation of 6x-5 changes our parabola.
As can be seen
the x-intercepts are the x-intercepts of the lines x+2 and 6x-5.
Also the parabola has shifted up the y-axis and it is not quite
as wide as (x+4)(6x-5).
Now how do you think changing the slope will affect the parabola?
Again the x-intercepts
correspond to the lines multiplied together to get the parabola.
But how is it different from the other 2 parabolas formed?
Just a reminder of the equations formed by the multiplication:
The x-intercepts for 6x2+19x-20 are -4 and 5/6. You would find this by setting the equation equal to zero and solving for x. Because it is a second degree equation there will be two values for x. So find the x-intercepts for the next two lines. After working the equations, the intercepts for 6x2+7x-10 are -2 and 5/6. And the intercepts for 12x2+14x-20 are -2 and 5/6 again. Notice that the last two lines have the same x-intercepts but they do not have the same shape. 12x2+14x-20 is a much longer and somewhat narrower parabola than 6x2+7x-10. These parabolas show that the shapes of the graphs can be different even though the x-intercepts are the same.
Now we will use composition on the same equations to see what kinds of graphs we will get. First we will compose in one direction then we will compose in the other direction.
Do you all think
that we will get a straight line, a parabola or something that
we have not seen yet? Well let's compose the first two equations
that we looked at and see.
Equation #1 - y=x+4
Equation #2 - y=6x-5
When you compose one equation with another you basically substitute one equation into another equation for the unknown element. So if we compose equation 2 into equation 1 it will look like:
Y= (6x-5) + 4
Then it needs to be worked out.
When we compose it with equation one inside equation two we get;
Y = 6(x+4)-5
Y = 6x + 24 -5
Y = 6x + 19
So looking at the equations, it shows that we still have linear equations. But the compositions produce different linear equations. The graphs of each line are shown below. Notice the slope and intercepts for each line. Can you guess what they will be before you find them on the graph?
Equation #1 - y=x+4
Equation #2 - y=6x-5
Equation #3 - y= 6x-1
Equation #4 - y = 6x + 19
The only one with a slope other than 6/1 is the first equation of y=x+4. All of the other lines have a slope of 6 but they have been shifted along the x-axis.
Now we can compose the other two equations with y=6x-5 and see what kind of graphs we get.
Equation #1 - y=x+2
Equation #2 - y=6x-5
Equation #3 - y= (6x-1)+2 = 6x + 1
Equation #4 - y = 6(x+2)-5 = 6x+12-5 = 6x + 7
The graphs for these look much like the last set of graphs with the slope staying the same in all but the y=x+2 equation.
We have now explored several different ways that two equations can relate to one another. The addition of two equations will change the slope as well as its position on the y-axis. But the resulting equation is still linear. The multiplication of two equations changes the graph from linear to a parabola. It illustrates how when an equation is quadratic it intersects the y-axis twice. When composition is used on the two equations, the resulting equation is still linear and the slope is the same as the slope of the composing equation. The use of the graphing calculator has enabled us to see this.