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Section 3.4. Slope and Rates of Change. Page 190. Slope. The rise , or change in y, is y 2  y 1 , and the run , or change in x, is x 2 – x 1. Example. Page 191. Use the two points to find the slope of the line. Interpret the slope in terms of rise and run. Solution. ( –4 , 1 ).

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section 3 4
Section 3.4
  • Slope and Rates of Change
slope

Page 190

Slope
  • The rise, or change in y, is y2y1, and the run, or change in x, is x2 – x1.
example
Example

Page 191

  • Use the two points to find the slope of the line. Interpret the slope in terms of rise and run.
  • Solution

(–4, 1)

(0, –2)

The rise is 3 units and the run is –4 units.

example1
Example

Page 192

  • Calculate the slope of the line passing through each pair of points.
  • a. (3, 3), (0, 4) b. (3, 4), (3, 2)
  • c. (2, 4), (2, 4) d. (4, 5), (4, 2)
  • Solution
example2
Example

Page 192

  • Calculate the slope of the line passing through each pair of points.
  • a. (3, 3), (0, 4) b. (3, 4), (3, 2)
  • c. (2, 4), (2, 4) d. (4, 5), (4, 2)
  • Solution
example3
Example

Page 192

  • Calculate the slope of the line passing through each pair of points.
  • a. (3, 3), (0, 4) b. (3, 4), (3, 2)
  • c. (2, 4), (2, 4) d. (4, 5), (4, 2)
  • Solution
example4
Example

Page 192

  • Calculate the slope of the line passing through each pair of points.
  • a. (3, 3), (0, 4) b. (3, 4), (3, 2)
  • c. (2, 4), (2, 4) d. (4, 5), (4, 2)
  • Solution
find the slope of the line containing the points 3 4 and 4 2

Finding Slope of a Line, p 249

Find the slope of the line containing the points (-3, 4) and (-4,- 2)

Find the slope of the line containing the points (4,-2) and (-1,5)

slope1

Page 193

Slope

Positive slope: rises from left to right

Negative slope: falls from left to right

slope2

Page 193

Slope

Zero slope:horizontal line

Undefined slope: vertical line

example5
Example

Page 193

  • Find the slope of each line.
  • a. b.
  • Solution
  • a. The graph rises 2 units for each unit of run m = 2/1 = 2.
  • b. The line is vertical, so the slope is undefined.
example6
Example

Page 193

  • Sketch a line passing through the point (1, 2) and having slope 3/4.
  • Solution
  • Start by plotting (1, 2).
  • The slope is ¾ which means a rise (increase) of 3 and a run (horizontal) of 4.
  • The line passes through the point (1 + 4, 2 + 3) = (5, 5).
slope as a rate of change

Page 195

Slope as a Rate of Change

When lines are used to model physical quantities in applications, their slopes provide important information.

Slope measures the rate of change in a quantity.

example7
Example

Page 195similar to Example 7&8and #87 from homeworkand #91

  • The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below.
  • a. Find the y-intercept. What does the y-intercept represent?
  • Solution
  • a. The y-intercept is 35, so the boat is initially 35 miles from the dock.
example cont
Example (cont)

Page 195similar to Example 7&8and #87 from homeworkand #91

  • The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below.
  • b. The graph passes through the point (4, 15). Discuss the meaning of this point.
  • Solution
  • b. The point (4, 15) means that after 4 hours the boat is 15 miles from the dock.
example cont1
Example (cont)

Page 195similar to Example 7&8and #87 from homeworkand #91

  • The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below.
  • c. Find the slope of the line. Interpret the slope as a rate of change.
  • Solution
  • c. The slope is –5. The slope means that the boat is going toward the dock at 5 miles per hour.
example 88 p 202
Example: #88 p 202
  • Electricity: The graph shows how voltage is related to amperage in an electrical circuit. The slope corresponds to the resistance in ohms. Find the resistance in this electrical circuit.
  • Find the slope of the line passing through the points. Look at the graph on page 202 and identify two points.
  • (0,0), (10, 20) and (20, 40) are possible
  • Interpret the slope as resistance in this electrical circuit.
  • 0.5 ohm
example 91 p 202
Example: #91 p 202
  • Median Household Income: In 2000, median family income was about $42,000, and in 2008 it was about $50,000.
  • Find the slope of the line passing through the points (2000,42000) and (2008,50000)
  • Interpret the slope as rate of change.
  • Median family income increased on average by $1000/year over this time period
    • If this trend continues, estimate the median family income in 2014.

$56,000 ($14000 added to $42000 or $6000 added to $50,000

example8
Example
  • When a street vendor sells 40 tacos, his profit is $24, and when he sells 75 tacos, his profit is $66.
  • a. Find the slope of the line passing through the points (40, 24) and (75, 66)
  • b. Interpret the slope as a rate of change.
  • Solution
  • b. Profit increases on average, by $1.20 for each additional taco sold.
example 86 on page 202
Example #86 on page 202
  • Profit from Tablet Computers: When a company manufactures 500 tablet computers, its profit is $100,000, and when it manufactures 1500 tablet computers, its profit is $400,000.
  • Find the slope of the line passing through the points (500, 100000) and (1500, 400000)
  • b. Interpret the slope as a rate of change.
  • The average profit is $300/tablets computer.
objectives
Objectives
  • Finding Slopes of Lines
  • Slope as a Rate of Change