# graphing equations in slope intercept form ... graphing equations in slope intercept form...

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Graphing Equations in Slope Intercept Form

Let’s remember a linear equation can be written in:

Standard Form: Ax + By = C

It can also be written as:

Slope-Intercept Form: y = mx + b (m = slope and b = y-intercept)

To Graph:

1. Plot the y-intercept (b in the equation).

2. From the y-intercept, use the slope to tell us how to move:

Example: m= → means from y-int. move UP 3 units and RIGHT 4 units.

m= → means from y-int. move DOWN 2 units and RIGHT 3 units

It is much easier to graph a line when the equation is in Slope-Intercept form.

Write an equation in slope-intercept form for a line with a slope of and a y-

intercept of -2. Then graph.

Plug in what you know:

y = mx + b

𝑦 = 𝑥 − 2

y-intercept = -2

Slope means up 3 right 4

Write an equation in slope-intercept form for the line shown in the graph.

To write an equation in slope-intercept form we need the slope and y-intercept.

(Hint: you will need to use the slope formula and two points on the line to

calculate slope).

In the above graph we see that the y-intercept = 1 or (0, 1).

Pick another point on the line (in this case, (2,0)).

Calculate slope using these two points: 𝑚 = 0−1

2−0 =

−1

2

Put it all together:

𝑦 = − 1

2 𝑥 + 1

EXAMPLE: The band boosters are selling sandwiches for $5 each. They bought

$1160 worth of ingredients. Write an equation for the profit made on x

sandwiches and graph.

𝑦 =5𝑥−1160

It’s important to know that x cannot be negative, since x represents number of sandwiches.

What would the profit be if 1400 sandwiches are sold (x=1400).

𝑦=5(1400)−1160

𝑦 =7000−1160

𝑦 =5840

The profit from selling 1400 sandwiches would be $5,840.

- 1160

Writing Equations in Slope-Intercept Form

Slope Intercept Form: y=mx + b (m = slope, b = y-int)

If we are given:

We can:

Slope and coordinate

A line has a slope of 1

2 and

passes through the point

(2, 4)

We can use the slope and coordinate to find y-int. (b):

𝑦 =𝑚𝑥+𝑏 Substitute m, x and y (from coordinate) and solve for b.

4=(2)( 1

2 )+𝑏

4=1+𝑏

3=𝑏

Now write the equation:

𝑦 = 1

2 𝑥+3

Two coordinate points

A line passes through the

points (1, 2) and (3, 5)

1. Calculate slope using slope formula

𝑚 = 5 − 2

3 − 1 =

3

2

2. Now that we have the slope, we can use the procedure from above to find b. Just use one of the given points:

2=(1)( 3

2 )+𝑏

1

2 = 𝑏

3. Write the equation 𝑦 = 3 2

𝑥 + 1

2

You Try! Write equations in slope-intercept form using the given information:

Slope = -

2

3 and passes through (3, 1)

𝑦 = − 2

3 𝑥 + 𝑏

1 = − 2

3 (3) + 𝑏

1 = −2 + 𝑏

3 = 𝑏

𝑦 = − 2

3 𝑥 + 3

Slope = 5 and passes through (0, 0)

𝑦 = 5𝑥 + 𝑏

0 = 5(0) + 𝑏

0 = 𝑏

𝑦 = 5𝑥

Line that passes through (-3, 1) and (4, 5)

𝑚 = 𝑦2 − 𝑦1 𝑥2 − 𝑥1

𝑚 =

5 − 1

4 + 3 =

4

7

1 = 4

7 (−3) + 𝑏

1 = −12

7 + 𝑏

2 5

7 = 𝑏

𝑦 =

4

7 𝑥 + 2

5

7

Line that passes through (0, -4) and (5, 5)

𝑚 = 𝑦2 − 𝑦1 𝑥2 − 𝑥1

𝑚 =

5 + 4

5 − 0 =

9

5

−4 = 9

5 (0) + 𝑏

−4 = 𝑏

𝑦 = 9

5 𝑥 − 4

I can write linear equations in slope-intercept form given…..

The slope and y-intercept

Slope = 𝟏

𝟑 y-int = -5

𝒚 = 𝟏

𝟑 𝒙 − 𝟓

Standard Form

4x – 2y = 14

𝟒𝒙 − 𝟐𝒚 = 𝟏𝟒

-𝟒𝒙 − 𝟒𝒙 −𝟐𝒚 = −𝟒𝒙 + 𝟏𝟒

𝒚 = 𝟐𝒙 − 𝟕

A graph

m = −𝟐

𝟓 b = -1

𝒚 =

−𝟐

𝟓 𝒙 − 𝟏

A point and a slope

(-1, 3) and slope = -3

𝒚 = 𝒎𝒙 + 𝒃

𝟑 = −𝟑(−𝟏) + 𝒃 𝟑 = 𝟑 + 𝒃

𝒃 = 𝟎 𝒚 = −𝟑𝒙

Two points

(-4, -7) and (8, -13)

𝒎 = −𝟕 + 𝟏𝟑

−𝟒 − 𝟖 =

𝟔

−𝟏𝟐 =

−𝟏

𝟐

𝒚 = 𝒎𝒙 + 𝒃

−𝟕 = − 𝟏

𝟐 (−𝟒) + 𝒃

−𝟕 = 𝟐 + 𝒃 𝒃 = −𝟗

𝒚 = − 𝟏

𝟐 𝒙 − 𝟗

Writing Equations in Point-Slope Form

Remember:

We’ve learned how to write linear equations in:

Standard Form: Ax + By = C

And

Slope-Intercept Form: y = mx + b

Today let’s write equations in Point-Slope Form. We use this form

when we are given the slope of the line and one coordinate on the

line.

Point-Slope Form: 𝒚 − 𝒚𝟏 = 𝒎(𝒙 − 𝒙𝟏)

where 𝑥1 𝑎𝑛𝑑 𝑦1 are given coordinates

EXAMPLE:

Write an equation in point-slope form for the line that passes through (3, -

2) with a slope of .

𝒚 − (−𝟐) = 1

4 (𝒙 − 𝟑)

𝒚 + 𝟐 = 1

4 (𝒙 − 𝟑)

NOTE: If given two points, use slope formula to get slope. Then use

the slope and either of the given coordinates to write the equation

in point-slope form.

Converting linear equations from one form to another:

To convert from one form to another, use the rules of algebra to make

the equation reflect the new form.

To convert to Standard Form (Ax + By = C):

Goal is to get the x term and y term on one side of the equal sign and

the constant on the other. Remember the rules for A, B and C!

To convert to Slope-Intercept Form (y = mx + b):

Goal is to get the y alone on one side of the equal sign.

YOU TRY!

Convert to Standard Form:

𝑦 − 1 = − (𝑥 − 5)

𝑦 − 1 = −2

3 𝑥 +

10

3

2

3 𝑥 + 𝑦 =

13

3

2𝑥 + 3𝑦 = 13

Convert to Slope-Int Form:

𝑦 + 3 = (𝑥 + 1)

𝑦 + 3 = 3

2 𝑥 +

3

2

𝑦 = 3

2 𝑥 −

3

2

Convert to Standard Form:

𝑦 = 1

2 𝑥 − 3

− 1

2 𝑥 + 𝑦 = −3

𝑥 − 2𝑦 = 6

Convert to Slope-Int Form:

3𝑥 + 4𝑦 = 12

4𝑦 = −3𝑥 + 12

𝑦 = − 3

4 𝑥 + 3

Parallel and Perpendicular Lines

Parallel lines: lines that never meet.

Perpendicular Lines: lines that cross and form right angles

NOTE: Since both of these consist of two lines, there are two equations.

Lines Rule Examples

Parallel

• The two lines have the SAME slope.

• The two lines have DIFFERENT y – intercepts.

𝑦=2𝑥+4 𝑦=2𝑥−3 𝑦=2𝑥

All have slope of 2

Perpendicular

• The two lines have slopes

that are OPPOSITE RECIPROCALS.

• Any y-intercepts.

𝑦=−3𝑥+6

𝑦 = 1

3 𝑥 − 2

-3 and opposite reciprocals

Intersecting

• There is NO relationship

between the slopes.

• Any y-intercepts.

𝑦=4𝑥+2 𝑦=−4𝑥−5

4 and -4 are not the same and they are not opposite reciprocals

It is easiest to work with equations in slope-intercept form

Steps in determining parallel and perpendicular lines or writing

equations of parallel and perpendicular lines:

Step 1: Convert the equation of the given line to slope-int form.

Step 2: Determine slope of the new line.

Step 3: Plug in what you know in y = mx + b to find the value of

b Step 4: Write the equation of the new line.

YOU TRY! Write an equation for the line that passes through (-3, 5) and is parallel to y = 2x – 4.

Slope of given line = 2 therefore

The slope of the second line = 2 (parallel- same) Find y-int (b) by using slope and coordinate:

5 = 2(−3) + 𝑏 5 = −6 + 𝑏

11 = 𝑏 Now write equation: