2-1 Slope-Intercept Form

2-1
Slope-Intercept Form

CommonCoreStateStandards HSA.CED.A.2,HSA.SSE.A.1,HSS.ID.C.7,MP.3,MP.4,MP.7

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Topic 2 Lesson 1

Slope-Intercept Form

I CAN…
write and graph linear equations using slope-intercept form.

VOCABULARY
• y-intercept

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Topic 2 Lesson 1

Alani wants to buy a $360 bicycle. She is considering two payment options.
The image shows Option A, which consists of making an initial down
payment then smaller, equal-sized weekly payments. Option B consists of
making 6 equal payments over 6 weeks.

A.

What factors should Alani take into consideration before deciding between
Option A and Option B?

B.

Communicate Precisely Suppose Alani could modify Option A and still pay off
the bike in 5 weeks. Describe the relationship between the down payment and
the weekly payments.

MODEL & DISCUSS

Open Ended

1. ​1. What factors should Alani take into consideration before deciding between Option A and Option B?

1. 2. Suppose Alani could modify Option A and still pay off the bike in 5 weeks. Describe the relationship between the down payment and the weekly payments.

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Open Ended

Define slope in your own words.

(In other words, what does the slope of a line tell you?)

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Match

Fill in the missing information

• $$m=\ \ \ \ \ \ \ \frac{ }{run}$$

• $$m=\frac{vertical\ change}{ }$$

• $$m=\frac{y_2-y_1}{ }$$

rise

horizontal change

• rise

• horizontal change

• $$x_2-x_1$$

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Topic 2 Lesson 1

ESSENTIAL QUESTION

What information does the slope-intercept form of a linear equation reveal about
a line?

Understand Slope-Intercept Form

How can you use slope-intercept form to graph linear equations?

Consider the equation . The equation is in slope-intercept form, y = mx + b , where m
is the slope of the line and b is the y -intercept.

Step 1 Identify the slope and y -intercept.

y = mx + b

EXAMPLE 1

CONCEPTUAL UNDERSTANDING

Step 2 Use the y -intercept to plot a point on the line.

The y -intercept is 2, so plot the point (0, 2) .

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Topic 2 Lesson 1

Understand Slope-Intercept Form

How can you use slope-intercept form to graph linear equations?

Consider the equation . The equation is in slope-intercept form, y = mx + b ,
where m is the slope of the line and b is the y -intercept.

Step 3 Use the slope to plot a second point.

Start at (0, 2) , move 4 units up and 5 units to the right to
locate a second point. Plot the point (5, 6) .

Step 4 Draw a line through the points.

EXAMPLE 1

USE STRUCTURE
Why is the point 4 units down and 5 units the left of (0, 2) also on the graph of

CONCEPTUAL UNDERSTANDING

Open Ended

Why is the point 4 units down and 5 units the left of ​(0, 2)​ also on the graph of

$$y=\frac{4}{5}x+2$$ ?

Graphing

Sketch the graph of $$y=\frac{3}{4}x-5$$

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Topic 2 Lesson 1

Write an Equation from a Graph

What is the equation of the line in slope-intercept form?

Step 1 Find the slope between two points on the line.

The line passes through (0, 1) and (4, −2).

Step 2 Find the y-intercept.

The line intersects the y-axis at (0, 1), so the
y-intercept is 1.

Step 3 Write the equation in slope-intercept form.

y = mx + b

The equation of the line in slope-intercept form is

EXAMPLE 2

COMMON ERROR
Remember that subscripts 1 and 2 in the slope formula indicate the y -values and the x -values are subtracted in the same order. It is important maintain the order y - and x -coordinates when you calculate the slope.

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Math Response

What is the equation of the line in slope-intercept form?

Type answer here

Correct Answer: y=\frac{2}{3}x-2

Deg^°

Rad

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Topic 2 Lesson 1

Write an Equation Given Two Points

What is the line that passes through the points (−1, −2) and (3, 4) ?

Step 1 Find the slope of the line.

Step 2 Use the slope and one point to find the y-intercept.

Step 3 Use the slope and the y-intercept to write the equation.

The equation in slope-intercept form of the line that passes through (−1, −2)

and (3, 4) is

EXAMPLE 3

STUDY TIP
It does not matter which point you designate as ( x1 , y1 ) and which is ( x2 , y2 ) . The slope will be same either way.

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Math Response

Write the equation in slope-intercept form of the line that passes through the points
(5, 4) and (−1, 6) .

Type answer here

Correct Answer: y=\frac{1}{3}x+\frac{17}{3}

Deg^°

Rad

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Topic 2 Lesson 1

Interpret Slope and y-Intercept

Amari takes the subway to his summer job and
back home every day. The cost of riding the
subway is the same every day in both directions.

A. Write an equation to represent the balance

on the fare card.

After each ride, the balance on Amari’s fare card decreases
by the cost of the ride. You can use a linear equation to
describe the relationship between the number of rides
taken x , and the balance on the fare card y .

Step 1 Find the slope.

Calculate the slope between the points (1,45.15) and (7,28.65) .

Step 2 Substitute the slope and the x - and y -coordinates of one
point into y = mx + b to find the y -intercept b .

y = mx + b

45.15 = −2.75(1) + b

47.9 = b

The equation y = −2.75x + 47.9 represents the balance on the fare card after x rides.

EXAMPLE 4

APPLICATION

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Topic 2 Lesson 1

Interpret Slope and y-Intercept

Amari takes the subway to his summer job and
back home every day. The cost of riding the
subway is the same every day in both directions.

B.What do the slope and y -intercept mean in
terms of the situation?
The slope of −2.75 means that the amount
of money on the fare card y decreases
by $2.75 for every ride that Amari takes.
The slope represents the cost of a subway ride.

The y -intercept 47.9 represents the balance on
the fare card before Amari has paid for any rides, $47.90 .

C.How many times can Amari ride the subway before
he needs to add money on his fare card?

Find the number of rides it takes for there to be $0 balance on the fare card.

y = −2.75x + 47.9

0 = −2.75x + 47.9

17.42 ≈ x

Amari can take 17 subway rides before he needs
to add money on the fare card.

EXAMPLE 4

REASON
Connect the solution of the equation with the subway context. Why does the solution 17.42 differ from the answer of 17 rides?

HAVE A GROWTH MINDSET
In what ways do you give your best effort and persist?

APPLICATION

Open Ended

REASON: Why does the solution 17.42 differ from the answer of 17 rides?

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Topic 2 Lesson 1

Interpret Slope and y-Intercept

Try It!

4.

Allie has a gift card for a coffee shop. After 2 coffees, the balance on the card is $19.70 ,
and after 4 coffees the balance is $13.40 . Write an equation that represents the balance
on the gift card, and interpret the parts of the equation.

EXAMPLE 4

Open Ended

Try It!

4. ​Allie has a gift card for a coffee shop. After 2 coffees, the balance on the card is ​$19.70​, and after 4 coffees the balance is ​$13.40​.

6. Write an equation that represents the balance on the gift card, and interpret the parts of the equation.

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Topic 2 Lesson 1

CONCEPT SUMMARY

Slope-Intercept Form of a Linear Equation

WORDS

The slope-intercept form of a linear equation reveals the slope and the y -intercept of
the line it represents.

ALGEBRA

NUMBERS

GRAPH

y = −2x − 1

The line has a slope of −2.
The y-intercept is −1.

The line has a slope of
The y-intercept is 1.

Open Ended

Do You UNDERSTAND?

1. ESSENTIAL QUESTION:  What information does the slope-intercept form of a linear equation reveal about a line?

Open Ended

Do You UNDERSTAND?

1. Communicate Precisely: How are the graphs of y = 2x + 1 and y = −2x + 1 similar?
   How are they different?

Open Ended

Do You UNDERSTAND?

Error Analysis: To graph $$y=\frac{2}{3}x+4$$ Emaan plots one point at (0, 4) and a second point

2 units right and 3 units up at (2, 7). He then draws a line through (0, 4) and (2, 7). What error did Emaan make?

Open Ended

4. Do You UNDERSTAND?

5. Make Sense and Persevere: When writing the equation of a line in slope-intercept form, how can you determine the value of m in y = mx + b if you know the coordinates of two points on the line?

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Topic 2 Lesson 1

Do You KNOW HOW?

Sketch the graph of each equation.

5.

y = 2x − 5

6.

Identify the slope and y-intercept of the line for each equation.

7.

8.

Write the equation of each line in slope-intercept form.

9.

11.

Aline that passes through (3, 1) and (0, −3)

12.

A line that passes through (−1, −5) and (4, −2)

10.

Graphing

Sketch the graph of

​y = 2x − 5

Graphing

Sketch the graph of

$$y=-\frac{3}{4}x+2$$

Open Ended

Identify the slope and y-intercept of the line

$$y=-5x-\frac{3}{4}$$

Open Ended

Identify the slope and y-intercept of the line

$$y=\frac{1}{4}x+5$$

Math Response

Write the equation of the line in slope-intercept form.

Type answer here

Correct Answer: y=-2x+1

Deg^°

Rad

Math Response

Write the equation of the line in slope-intercept form.

Type answer here

Correct Answer: y=\frac{3}{2}x-2

Deg^°

Rad

Math Response

Write the equation of the line in slope-intercept form.
A line that passes through (3, 1) and (0, −3)

Type answer here

Correct Answer: y=\frac{4}{3}x-3

Deg^°

Rad

Math Response

Write the equation of the line in slope-intercept form.
A line that passes through (−1, −5) and (4, −2)

Type answer here

Correct Answer: y=.6x-4.4

Deg^°

Rad

Quiz