graphing equations in slope intercept form...graphing equations in slope intercept form let’s...

of 16 /16
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 = −2 y-intercept = -2 Slope means up 3 right 4

Upload: others

Post on 23-Apr-2020

4 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

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

Page 2: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

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

Page 3: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

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

Page 4: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

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

2𝑥+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

2𝑥 +

1

2

Page 5: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

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

Slope = -

2

3 and passes through (3, 1)

𝑦 = −2

3𝑥 + 𝑏

1 = −2

3(3) + 𝑏

1 = −2 + 𝑏

3 = 𝑏

𝑦 = −2

3𝑥 + 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

7𝑥 + 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

5𝑥 − 4

Page 6: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

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

The slope and y-intercept

Slope = 𝟏

𝟑 y-int = -5

𝒚 = 𝟏

𝟑𝒙 − 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

𝟓𝒙 − 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

𝟐𝒙 − 𝟗

Page 7: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

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.

Page 8: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

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

3𝑥 +

10

3

2

3𝑥 + 𝑦 =

13

3

2𝑥 + 3𝑦 = 13

Convert to Slope-Int Form:

𝑦 + 3 = (𝑥 + 1)

𝑦 + 3 =3

2𝑥 +

3

2

𝑦 =3

2𝑥 −

3

2

Convert to Standard Form:

𝑦 =1

2 𝑥 − 3

−1

2𝑥 + 𝑦 = −3

𝑥 − 2𝑦 = 6

Convert to Slope-Int Form:

3𝑥 + 4𝑦 = 12

4𝑦 = −3𝑥 + 12

𝑦 = −3

4𝑥 + 3

Page 9: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

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

3𝑥 − 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

Page 10: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

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:

𝑦 = 2𝑥 + 11

Write the equation of the line that passes through (-4, 6) and is perpendicular to 2x + 3y = 12. First we need to put the first equation in slope intercept form:

3𝑦 = −2𝑥 + 12

𝑦 = −2

3𝑥 + 4

Now we can see that the slope for the

perpendicular line is 3

2 (opposite reciprocal

of −2

3 )

Find y-int by using slope and coordinate:

6 = (−4)(3

2) + 𝑏

6 = −6 + 𝑏 12 = 𝑏

Now write equation: 𝑦 = 3

2𝑥 + 12

Page 11: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

We can prove two or more lines are parallel or perpendicular to each

other just by looking at the slopes of the lines.

Determine whether these lines are parallel or perpendicular to each

other:

Looking at the slopes we see that Line 2 and Line 3 are parallel,

Line 1 and Line 2 are perpendicular and

Line 1 and Line 3 are perpendicular

Line 1: 3x + y = 12

Line 2: y = 1

3 x + 2

Line 3: 2x – y = 6 - 5

Line1:

3 𝑥 + 𝑦 = 12

𝑦 = − 3 𝑥 + 12

𝑠𝑙𝑜𝑝𝑒 = − 3

Line 2:

𝑦 = 1

3 𝑥 + 2

𝑠𝑙𝑜𝑝𝑒 = 1

3

Line 3:

2 𝑥 − 6 𝑦 = − 5

− 6 𝑦 = − 2 𝑥 − 5

𝑦 = − 2

− 6 𝑥 −

5

6

𝑦 = 1

3 𝑥 −

5

6

𝑠𝑙𝑜𝑝𝑒 = 1

3

First step is to

convert all to

slope - int form

Page 12: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

Parallel and Perpendicular

Use the graph to answer the following two questions:

A (2, 4) B(3, 6) C (6, 4) D (5, 2)

1. Is angle B a right angle? METHOD TO SOLVE: To answer this question, we need to determine if line AB is perpendicular to line BC. Use the slope formula to find slope of both lines – if they are opposite reciprocals, the we have perpendicular lines which makes angle B a right angle.

2. Is each pair of opposite sides

parallel? METHOD TO SOLVE: Find the slope of all four lines using slope formula. If opposite sides have equal slopes then they are parallel.

A

B

C

D

Page 13: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

Scatter Plots and Line of Best Fit

Bivariate data Data with two variables

Scatter Plot Shows the relationship between a set of data

with two variables. The data is graphed as

ordered pairs on a coordinate plane.

Line of fit (trend) Line that models a trend within the data.

Examples of Bivariate Data:

The number of hours worked and wage earned

The number of hours studying and grades.

Can you name some other examples?

Scatter Plots:

Positive slope Negative Slope No

Positive Correlation Negative Correlation Correlation

Page 14: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

We can draw and write an equation for a line that shows the trend

of the data ( Line of Best Fit ).

Step 1: Make scatter plot & determine relationship

Step 2: Draw a line that passes close to most of the data

Step 3: Use two points on the line you drew (not

necessarily part of the data) to write an equation.

Step 4: Use that line to make predictions.

I will use points (16, 300) and (22, 500):

𝑚 =500−300

22−16=

200

6=

100

3

𝑦 = 𝑚𝑥 + 𝑏

300 = (16) + 𝑏

300 = 533.33 + 𝑏

−233.33 = 𝑏

Line of fit equation is: 𝑦 = 𝑥 − 233.33

Page 15: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

Inverse Linear Functions

Inverse Relation The set of ordered pairs that we get by

switching the x-coordinate and y-coordinate.

Inverse Relations

Relation A Relation B

(-3, -16) (-16, -3)

(-1, 4) (4, -1)

(2, 14) (14, 2)

(5, 32) (32, 5)

YOU TRY:

Identify the Inverse Relation:

Relation: (3, 5), (4, 6), (5, 7)

Inverse: (5, 3), (6, 4), (7, 5)

Relation: (-1, -1), (2, 3), (4, 7)

Inverse: (-1, -1), (3, 2), (7, 4)

Relation: (.1, 5), (.2, 6), (.3, 7)

Inverse: (5, .1), (6, .2), (7, .3)

Relation: (8, 4), (6, 3), (4, 2)

Inverse: (4, 8), (3, 6), (2, 4)

Switch x and y values

Page 16: Graphing Equations in Slope Intercept Form...Graphing Equations in Slope Intercept Form Let’s remember a linear equation can be written in: Standard Form: Ax + By = C It can also

Function Symbols: f(x) read as function of x (the output)

𝒇−𝟏(𝒙) read as inverse of f(x)

Steps to finding the inverse of a function:

Step 1: Replace f(x) with y

Step 2: Switch x and y

Step 3: Isolate y (get y alone)

Step 4: Replace y with 𝑓−1(𝑥)

YOU TRY! a) f(x) = 4x - 8 b) f(x) = x + 11

Step 1:

Step 2:

Step 3:

Step 4:

𝑦 = 4𝑥 − 8

𝑥 = 4𝑦 − 8

4𝑦 = 𝑥 + 8 𝑦

= 𝑥 + 2

1

𝑓−1 = 𝑥 + 2

4

𝑦 = − 𝑥 + 11

𝑥 = − 𝑦 + 11

𝑦 = −𝑥 + 11

𝑦 = −2𝑥 + 22

𝑓−1 = −2𝑥 + 22