6-3: Linear Reasoning – Slope and Intercept

The naming of addresses in Utah was originally based on the idea of a coordinate plane. In Orem, there are two major roads that go through and intersect at the center of the city. The street that goes north and south is called Main Street. The street that goes east and west is called Center Street. (In reality, Main Street and Center Street never actually meet in Orem!)

Any address that lies north of Center Street is regarded as being on the north side of the city. Any address that lies south of Center Street is regarded as being on the south side of the city. Any address lying east of Main Street is regarded as being on the east side of the city. Any address that lies west of Main Street is regarded as being on the west side of the city.

An address uses two numbers to show where it lies in the city. Each number is called a coordinate. The first coordinate is the horizontal east/west position and the second coordinate is the vertical north/south position. Identify the following addresses on the map.

 

 

 

1. 1. 1. 1. (100 E, 400 N)
         2. (100 W, 300 N)
         3. (400 W, 100 S)
         4. (300 E, 500 S)

 

 

 

Solution

1. To get to 100 E, we start at the origin (the intersection of Main Street and Center Street) and move east (right) to 100. Then to get to 400 N, we turn north (up) and move to the level of 400. We are now at (100 E, 400 N).
2. To get to 100 W, we start at the origin (the intersection of Main Street and Center Street) and move west (left) to 100. Then to get to 300 N, we turn north (up) and move to the level of 300. We are now at (100 W, 300 N).
3. To get to 400 W, we start at the origin (the intersection of Main Street and Center Street) and move west (left) to 400. Then to get to 100 S, we turn south (down) and move to the level of 100. We are now at (400 W, 100 S).
4. To get to 300 E, we start at the origin (the intersection of Main Street and Center Street) and move east (right) to 300. Then to get to 500 S, we turn south (down) and move to the level of 500. We are now at (300 E, 500 S).

Based on Explore 1, we may change the name “Center Street” to “-axis”, and the name “Main Street” to “-axis”. We may also consider “East” or “North” with “+” (positive) and “West” or “South” with “–” (negative). By doing so, you have (+, +) address coordinates at the east and north of the city (also called the first quadrant), (–, +) address coordinates at the west and north of the city (also called the second quadrant), (–, –) address coordinates at the west and south of the city (also called the third quadrant), and (+, –) address coordinates at the east and south of the city (also called the fourth quadrant). This means that the -axis and -axis are number lines that meet at and . Consequently, the origin is the point (0, 0). Show the following addresses on the map.

 

 

1. (+400, +200)
2. (–300, +400)
3. (–400, –300)
4. (+500, –500)

 

 

 

 

Solution

 

1. We start at the origin (0, 0) and move right to +400, then go up to the level of +200. We are now at (+400,+200).
2. We start at the origin (0, 0) and move left to –300, then go up to the level of +400. We are now at (–300, +400).
3. We start at the origin (0, 0) and move left to –400, then go down to the level of –300. We are now at (–400, –300).
4. We start at the origin (0, 0) and move right to +500, then go down to the level of –500. We are now at (+500, –500).

 

Identify the coordinates of each point on the coordinate plane. Use a pair of parentheses to enclose the -coordinate and the -coordinate. For example, (2, 3) means the location of a point where the point lies on a vertical line that passes through 2 on the -axis and on a horizontal line that passes through 3 on -axis.

Solutions

To determine the -coordinate of each point, we imagine a vertical line that passes through the point and determine where that line crosses the -axis. For point A, a vertical line moves down from the point and hits the -axis at 3.

To determine the -coordinate of each point, we imagine a horizontal line that passes through the point and determine where that line crosses the -axis. For point A, a horizontal line moves left from the point and hits the -axis at 5.

Consequently, point A has coordinates (3, 5).

A = (3, 5); B = (–8, 7); C = (8, 2); D = (–5, –2); E = (2, –8); F = (–3, –7); G = (-4, 2); H = (8, –5)

Given a table with two columns where the first column indicates the -coordinate and the second column indicates the -coordinate, transfer each row as a coordinate pair and identify the coordinate pair on the coordinate plane.

		Coordinates
2	7	(2, 7)
3	–4
–2	–1
–8	6
3	–7
7	4
–7	–5
–3	3

Solution

		Coordinates
2	7	(2, 7)
3	–4	(3, –4)
–2	–1	(–2, –1)
–8	6	(–8, 6)
3	–7	(3, –7)
7	4	(7, 4)
–7	–5	(–7, –5)
–3	3	(–3, 3)

The corner store has a shelf where everything on it is sold for $2 per item.

1.  Use this rate to create a two-column table where the first column has values of the independent variable (i.e., the number of items) from 0 to 6 and the second column has values of the dependent variable (i.e., cost).

Solution

Number of Items	Cost ($)
0	0
1	2
2	4
3	6
4	8
5	10
6	12

 

2. Suppose represents the number of items (independent variable) and represents the cost (dependent variable). Transfer each coordinate pair on each row of the table to the coordinate plane. Note that we usually place the independent variable on the -axis and the dependent variable on the -axis.

Solution

3. What does the point (0, 0) represent?

Solution

It shows that when 0 items are bought the cost is $0. This is the initial value of the linear pattern.

4. What do you notice about all the points on the coordinate plane?

Solution

They all lie on a line. They all have both – and -coordinates that are positive.

5. Explain how we would move from the point (1, 2) to (2, 4).

Solution

We could move 1 unit to the right horizontally, then move 2 units up vertically or we could move 2 units up vertically, then move 1 unit to the right horizontally.

6. Explain how we would move from the point (5, 10) to (6, 12).

Solution

We could move 1 unit to the right horizontally, then move 2 units up vertically or we could move 2 units up vertically, then move 1 unit to the right horizontally.

7. How do you explain the solutions to questions 4 and 5, and what do they have to do with the unit rate of $2 per item?

Solution.

The answers are the same. Moving 1 unit to the right horizontally results in an increase of 1 item. Moving 2 units up vertically results in a $2 increase in cost. So, the (vertical increase) ÷ (horizontal increase) = the unit rate.

We do not need to create a table of values showing a linear pattern and then transfer the results onto the coordinate plane to use linear reasoning. We can actually use linear reasoning on the coordinate plane directly.

Suppose the corner store has another shelf where everything on the shelf sells for $3 per item. This is a unit rate, so we know that the price increases by $3 as the number of items purchased increases by 1. So, if = the number of items (independent variable) and = the cost (dependent variable), we can start with any coordinate pair and use the unit rate to plot all other coordinate pairs.

We know that if we buy 1 item the cost will be $3, so the coordinate pair (1, 3) lies on the coordinate plane.

1. Plot the point (1,3).

Solution

2. Use the unit rate of $3 per item to move from the point (1, 3) to another point on the coordinate plane.

Solution

The -value increases by 3 as the -value increases by 1. Therefore, the next point after (1, 3) is (1 + 1, 3 + 3) = (2, 6).

3. Use this pattern to continue determining other points on the coordinate plane.

Solution

4. What pattern is formed using the unit rate and a starting point?

Solution

The unit rate tells us how to get from one point to the next. In this case the unit rate of $3 per item tells us to move horizontally right by 1 unit and 3 units vertically up to get to the next point. It is like climbing a staircase of 1 over 3 up every time. The points all lie on a line.

We call the number of units we move vertically to get from one point to another the rise. The number of units we move horizontally from one point to another is the run.

5. If we skip a point and move from (1, 3) to (3, 9) what is the ratio of the vertical rise to the horizontal run?

Solution

We move 2 units to the right, and 6 units up, so the ratio is 6/2 = 3/1.

6. If we skip 2 points and move from (1, 3) to (4, 12) what is the ratio of the vertical rise to the horizontal run?

Solution

We move 3 units to the right, and 9 units up, so the ratio is 9/3 = 3/1.

7. How do the answers to 6 and 7 relate to the unit rate in a linear pattern?

Solution

They are equal. The unit rate = rise/run.

8. How can we move from the point (1, 3) to the initial value of zero represented by the point (0, 0)? How does this relate to the unit rate of the linear pattern?

Solution

We have to go back down the staircase. So we would go down 3 units and left 1 unit. This means that the rise would be –3 since we are going down, not up. Also, the run would be –1, since we are going left, not right. So rise/run = –3/–1 = 3/1 = the unit rate.

A swimming pool loses 10 inches of water per month to evaporation over the summer. If the original depth of the water is 5 feet, the graph shows the linear pattern of the depth of water in the pool over time.

 

1. What is the unit of measurement on the vertical scale? What is the unit of measurement on the horizontal scale?

Solution

The vertical scale is measured in inches, and the horizontal scale is measured in months.

2. How do we know the plotted points represent a linear pattern?

Solution

The evaporation rate is constant at 10 inches per month. The points all lie on a line. 

3. What does the point (0, 60) represent?

Solution

(0, 60) tells us that at the start of the summer the water depth was 60 inches. It is the initial value of the linear pattern.

4. What is the rise and run from one point to the next point? Explain your reasoning.

Solution

The rise is –10 and the run is 1 from one point to the next point. For example, from the point (1, 50) to the point (2, 40), the vertical rise from 50 to 40 is 40 – 50 = –10. The horizontal run from 1 to 2 is 2 – 1 = 1.

5. What is the rate of change of the linear pattern? Write the rate in fraction form. Explain your reasoning.

Solution

The rate of change is because the depth decreases by 10 inches every month.

6. Do you agree that the rate of change of a linear pattern is equivalent to the fraction of rise over run on the graph? Explain your reasoning.

Solution

Yes, the rate is exactly the rise of –10 over the run 1 that is on the coordinate plane. Only the units of measurement are missing from the fraction rise/run. However, they are implied by the title of the axes.

 

Both the variables months and depth are continuous. This means that we do not jump from one month to the next, nor does the depth of the pool jump from one depth to another. Rather time flows by without taking a break, and the depth of the pool declines slowly over time. Consequently, the graph of discrete plotted points really should be continuous too. We can achieve this by simply joining the points with a line.

7. Turn your graph of plotted points into a continuous graph.

Solution

The points join to form a line that travels from (0, 60) to (6, 0). The point (0, 60) represents the initial value of a depth of 60 inches at the start of the summer. This is the point where the line intersects the vertical axis. Consequently, it is called the -intercept.

8. State the point that is the -intercept and explain what it means.

Solution

The -intercept is the point where the line intersects the -axis: (6, 0). This point tells us that at 6 months the pool is dry.

Ground beef is priced at $5.99 per pound at the local grocery store.

1. Create a table of values showing the linear pattern for the cost of buying 0 – 5 pounds of ground beef.
2. What variable is the independent variable and what are the measurement units?
3. What variable is the dependent variable and what are the measurement units?
4. What is the unit rate?
5. Transfer the data points to a coordinate plane, and draw the line that represents the linear pattern.
6. What is the -intercept of the line, and how does it relate to the initial value in the table?
7. What is the slope of the line and how does it relate to the rate in the table?
8. Use the graph to determine the cost of 2.5 pounds of ground beef?
9. Use the graph to determine how much ground beef can be bought for $20.

Solutions

1. Weight (lb)	Cost ($)
   0	0
   1	5.99
   2	11.98
   3	17.97
   4	23.96
   5	29.95

2. Independent variable is weight measured in pounds.
3. Dependent variable is cost measured in dollars.
4. Rate = $5.99/1 lb.
5. 

 

7. The -intercept is the point where the line intersects the -axis: (0, 0). The initial value = $0. So the initial value = the -coordinate of the -intercept.
8. To determine the slope, we choose any two points on the line and find the rise and the run. Between (2, 11.98) and (3, 17.97) the rise = 17.97 – 11.98 = 5.99, the run = 3 – 2 = 1. So the slope = rise/run = 5.99/1. The rate is $5.99/1 lb. The slope and the rate are identical.
9. At = 2.5, 15. It is very difficult, if not impossible to read decimal values from a graph (see dotted purple lines). But 2.5 lb of ground beef costs about $15. The exact value is $14.975.
10. When = 20, 3.4, so we can buy about $3.4 pounds of ground beef for $20 (see dotted red lines). The  value is actually 3.339 lb.

The graph shows three scenarios. Line 1 represents the cost of wild bird seed by weight in pounds with free shipping. Line 2 represents the sale of wild bird seed with a set shipping charge of $4. Line 3 represents the sale of wild bird seed with a set shipping charge of $10.

1. What are the -intercepts of each line and how do they relate to the shipping charges?

Solution

Line 1 intersects that -axis at (0, 0). The shipping charge is $0. Line 2 intersects that -axis at (0, 4). The shipping charge is $4. Line 3 intersects that -axis at (0, 10). The shipping charge is $10. the -coordinate of the -intercept equals the shipping charge.

2. What is the slope of each line and what do the slopes tell us?

Solution

Using points (0, 0) and (2, 6), the slope of line 1 = rise/run = (6 – 0)/(2 – 0) = 3.

Using points (0, 4) and (2, 10), the slope of line 1 = rise/run = (10 – 4)/(2 – 0) = 3.

Using points (0, 10) and (2, 16), the slope of line 1 = rise/run = (16 – 10)/(2 – 0) = 3.

All the slopes = 3. Since slope = rate of change, this tells us that the bird seed sells for $3 per pound.

Since all three lines have the same slope, they all go in the same direction. Such lines are called parallel lines.

The graph shows two different wages (dollars per hour).

1. What are the two wages?
2. What are the slopes of the two lines?
3. Which one is steeper?
4. How is steepness related to slopes?

Solutions

1. $2/1 hour, $3/1 hour
2. The slope of the blue line is 3/1 or 3, and the slope of the red line is 2/1 or 2.
3. The blue line.
4. The larger a slope is, the steeper it is.

A car’s value decreases each year.  The graph shows the value in thousands of dollars of two cars over time .

1. What are the -intercepts of each line and what do they tell us?

Solution

The line representing the worth of Car 1 intersects the -axis at (0, 40). This tells us that the initial value of Car 1 was $40,000. The line representing the worth of Car 2 intersects the -axis at (0, 32). This tells us that the initial value of Car 2 was $32,000.

2. What is the slope of each line and what do they tell us?

Solution

Using points (0, 40) and (4, 28) the slope of the line for Car 1 is rise/run = (28 – 40)/(4 – 0) = –12/4 = –3/1. This tells us that the value of Car 1 is decreasing at a rate of $3000 per year.  Using points (0, 32) and (4, 24) the slope of the line for Car 2 is rise/run = (24 – 32)/(4 – 0) = –8/4 = –2/1. This tells us that the value of Car 1 is decreasing at a rate of $2000 per year.

3. Where do the two lines intersect? What does this tell us?

Solution

The two lines intersect at the point (10, 16). This tells us that the two cars will have the same value of $16,000 when they are 10 years old.

4. Which line is steeper and how is this shown with their slopes?

Solution

The line representing Car 1 is steeper; it falls faster than the line representing Car 2. The slope for Car 1’s line is –3/1. The slope for Car 2’s line is –2/1.  The negative slopes tell us that the cars’ values are going down. Since 3 > 2, –3 is more negative than –2, so Car 1’s value is decreasing faster than Car 2’s value.