MA 0090 Section 22 - More on Graphing Lines
Wednesday, April 5, 2017
Objectives: Graphing using slope-intercept and standard form.
A few more examples using slope-intercept form
As we did it last time, to graph the line from a linear equation in slope-intercept form
(1)
y = mx + b,
we note that the graph will cross the y-axis at b, and then we can find a second point using the slope, m.
For example, consider the equation
2
x − 2.
3
up 2
The y-intercept is b = −2, and the slope is m = 23 =
. The y-intercept has coordinates (0, −2), and
right 3
counting 3 to the right and 2 up from the y-intercept gets us to the point (3, 0). We get the graph below.
(2)
y=
up 2
right 3
Now, consider the equation
1
y = − x + 2.
2
down 1
Here, the y-intercept is b = 2, and the slope is − 12 = −1
2 = right 2 . The graph comes out as follows.
(3)
right 2
down 1
1
MA 0090 Section 22 - More on Graphing Lines
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One last example. Consider the equation
(4)
y = 3x − 2.
We have a y-intercept of b = −2, and a slope of m = 3 =
3
1
=
up 3
.
right 1
up 3
right 1
Quiz 22, Part I
Graph the following lines, which are in slope-intercept form. In D2L, give the coordinates of the y-intercept,
and the second point found from the simplest interpretation of the slope.
1.
y = − 43 x + 2
2.
y = 2x − 4
3.
y = 32 x + 3
Graphing linear equations in standard form
Consider the linear equation
(5)
4x − 2y = 8.
When we have “x-term - y-term equals constant term,” we say that the equation is in standard form. In
general, linear equations in standard form look like
(6)
ax + by = c.
One way to graph a linear equation in standard form would be to convert it to slope-intercept form. We can
do that by adding and multiplying things to both sides, as we have done before, when we solved equations.
Here, we’re solving for y. Back to the original example, let’s put that equation into slope-intercept form.
(7)
4x − 2y = 8
4x − 2y − 4x = 8 − 4x
−2y = 8 − 4x
−2y
8 − 4x
=
−2
−2
y = −4 + 2x.
We would normally write this as y = 2x − 4. To graph it, we would have m = 2 =
2
1
=
up 2
and b = −4.
right 1
MA 0090 Section 22 - More on Graphing Lines
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up 2
right 1
A simpler alternative would be to find the two intercepts of the line, since it’s generally easier to find the
intercepts than it is to convert to slope-intercept form. Since the x-intercept has coordinates (∗∗, 0) and the
y-intercept has coordinates (0, ∗∗), we only have to substitute zeros into the equation
(8)
4x − 2y = 8
For the x-intercept, we have
(9)
4x − 2(0) = 8
4x = 8
x=2
Since y = 0, the y-term is equal to zero, and we can generally solve 4x = 8 in our heads. I would normally
just cover up the y-term, and see that x = 2.
For the y-intercept, I would cover up the x-term, and see that y = −4. Actually going through the steps,
we would get
4(0) − 2y = 8
(10)
−2y = 8
y = −4
We now have the two points we need to draw the graph, (2, 0) and (0, −4). Or equivalently, the line will
cross the x-axis at x = 2, and the y-axis at y = −4.
Here’s another example. Graph 3x + 4y = 12. Again, I would just cover up the y-term to see that x = 4 is
the x-intercept, and then cover up the x-term to see that the y-intercept is y = 3. The steps would look like
3x + 4(0) = 12
3x = 12
(11)
x = 4,
MA 0090 Section 22 - More on Graphing Lines
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and
3(0) + 4y = 12
4y = 12
(12)
y = 3.
The graph looks like
Quiz 22, Part II
Graph the linear equations, which are in standard form. In D2L, give the coordinates of the x- and yintercepts.
4.
x+y =3
5.
x − 2y = 8
6.
−2x + 3y = 12
Homework 22
Match the following linear equations with their graphs on the following page. One way to do this would be
to graph the line, and compare them with the graphs shown.
1.
y = 32 x + 2
2.
y = 4x − 3
3.
y = −3x + 3
4.
y = − 23 x + 2
5.
4x + 3y = 6
6.
−3x + y = 3
7.
8x − 2y = 6
8.
−2x + 3y = 6
MA 0090 Section 22 - More on Graphing Lines
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Graph A
Graph B
Graph C
Graph D
Graph E
Graph F
Choose answer G, if the correct graph is not shown.
Match the following linear equations in standard form with the linear equations in slope-intercept form. One
way to do this would be to convert the standard form to slope-intercept form as in the first example in the
second section.
9.
3x + 2y = 6
A.
y = −4x + 3
10.
−3x + 4y = −8
B.
y = 34 x − 2
11.
−5x + 2y = 2
C.
y = − 32 x + 3
12.
4x + y = 3
D.
y = 52 x + 1
13.
4x − 2y = 2
E.
y = −x + 5
14.
x+y =5
F.
y = 2x − 1
Choose answer G, if the correct equation is not shown.