UNIT 2Linear functionsLesson 2 of 31 Linear and Nonlinear Functions2 Rate of Change and Initial Value3 Describing Graphs: Increasing and Decreasing
Rate of Change and Initial Value
A linear model is an equation y = mx + b that fits a story. The slope is the rate. The y-intercept is the start.
What you'll be able to do
- Write y = mx + b from a story, two readings or a table.
- Say what m and b mean, with units.
- Use the model to predict, or to find when a value is reached.
- Tell when the model stops making sense.
The big idea
A model fits a story. It is an equation y = mx + b. x and y stand for amounts in the story.
The rate is m. It is how much y changes for each 1 of x. Say it with units, like $40 per hour.
The start is b. It is the value of y when x = 0. Say it with units too.
An example. A food truck opens with $150. It takes in $40 an hour. The model is y = 40x + 150.
From two readings. A car has 52 L at 2 hours and 36 L at 6 hours. The rate is (36 − 52) ÷ (6 − 2) = −4 L an hour.
Step back to the start. From 2 hours back to 0, add 2 × 4 = 8 L. So b = 60 and y = −4x + 60.
Use the model both ways. Put in x to predict y. Put in y and solve to find x.
Know where it stops. The car's fuel can't go below 0. The model only works from 0 to 15 hours.
Words you need
- Model
- An equation that fits a real situation.
- Rate
- How much y changes for each 1 of x. It is the slope.
- Initial value
- The value of y when x = 0. It is the y-intercept.
- Units
- What a number counts, like liters, dollars or hours.
- Range of a model
- The x-values where the model makes sense.
- Slope
- The rate m in y = mx + b: rise ÷ run.
Reading a model
| Part of y = mx + b | What it means | Example: y = 40x + 150 |
|---|---|---|
| m | Change in y for each 1 of x | $40 per hour |
| b | y when x = 0 | $150 at the start |
| Put in x | Predict y | After 3 hours: $270 |
| Set y and solve | Find when it is reached | $350 after 5 hours |
Worked examples
Example 1
A food truck has $150 in its register when it opens. It takes in $40 an hour. Write a model and say what each number means.
- The start is b = 150, the dollars at 0 hours.
- The rate is m = 40, the dollars per hour.
- So the model is y = 40x + 150.
Answer: y = 40x + 150
Example 2
A car has 52 L of fuel after 2 hours and 36 L after 6 hours. Write a model. When is it empty?
- The rate is (36 − 52) ÷ (6 − 2) = −4 L an hour.
- Step back 2 hours: 52 + 8 = 60. So b = 60.
- The model is y = −4x + 60.
- It is empty when −4x + 60 = 0, which is at 15 hours.
Answer: y = −4x + 60. It is empty after 15 hours.
Example 3
Graph y = 40x + 150 for 0 to 3 hours. The y-axis goes up $50 a square.
- Plot the start, (0, 150). That is 3 squares up.
- Go right 1 and up 40 to (1, 190).
- Then (2, 230) and (3, 270). Draw the line.
Answer: (0, 150), (1, 190), (2, 230), (3, 270)
Common mistakes
Swapping the rate and the start. The rate goes with x. A $25 fee and $18 a month is y = 18x + 25.
Dropping the units. Say '$2.75 per mile', not just '2.75'.
Using the first reading as the start. The start is y at x = 0. Step back from the first reading.
Using the model past its end. A balloon can't be below the ground. Stop where the model stops making sense.
Practice: 22 questions
21 of 22 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- A pool has 1,200 L of water. A hose adds 150 L a minute. Write a model for the water y after x minutes. What do 150 and 1,200 mean? Word problem
- A taxi ride costs y = 2.75x + 4 dollars for x miles. What does 2.75 mean? What does 4 mean? Word problem
- A plant is 3 cm tall on day 0 and 15 cm tall on day 6. It grows at a steady rate. Write a model for its height y after x days. Word problem
- A gym charges $25 to join and $18 a month. Ken writes y = 25x + 18 for the cost after x months. Find his mistake. Word problem
- A phone's charge is y = 96 − 6x percent after x hours. When does it reach 0%? Word problem
- A bike rental costs $8 plus $3 an hour. Write the model. Which points do you plot for 0, 1 and 2 hours? Word problem
Tier 2 Intermediate One more step to think through
- A printer prints 45 pages every 3 minutes. It has already printed 120 pages. Write a model for the pages y after x more minutes. Word problem
- A water heater is y = 18 + 4x degrees after x minutes. When does it reach 60 degrees? Word problem
- A graph of a road's height goes through (100, 40) and (300, 24). x is meters along the road. y is height in meters. Write a model. What does the slope mean? Word problem
- A balloon's height is y = 300 − 12x m after x seconds. It lands when its height is 0. Ade says it is at −60 m after 30 seconds. Find his mistake. Word problem
- A hiker is at 2,350 m after 1 hour, 2,050 m after 3 hours and 1,900 m after 4 hours. Write a model for her height. What do m and b mean? Word problem
- A savings jar has y = −25x + 200 dollars after x weeks. Which points do you plot for 0, 1 and 2 weeks? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- A town had 8,400 people in 2018 and 9,600 in 2022. It grows by the same number each year. Let x be years since 2018, so 2018 is x = 0 and 2022 is x = 4. Write the model. Word problem
- A snail is 50 cm from a wall. It crawls toward it at 6 cm every 4 minutes. An hour is 60 minutes. Write a model for its distance from the wall after x hours. Word problem
- A hall costs y = 35x + 200 dollars for x guests. The model is for 20 to 80 guests. Ana says an empty hall costs $200. Find her mistake. Word problem
- A spring is 12 cm long with a 2 kg weight. It is 15 cm long with a 4 kg weight. Write a model. What does the y-intercept mean? Can the spring ever be 0 cm long? Word problem
- A coach has $240 for basketballs at $30 and soccer balls at $20. The model is 30x + 20y = 240. Write it as y = mx + b. What do the slope and the y-intercept mean? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- A bakery pays $600 a month in rent. Each cake costs $8 to make and sells for $20. Write a model for the month's profit y after x cakes. Break even means a profit of 0. How many cakes to break even? Word problem
- A table should be linear: (1, 5), (2, 9), (4, 17), (7, 29), (10, 40). One y-value is wrong. Which one? What should it be?
- Studio A charges y = 15x + 40 dollars for x hours. Studio B charges y = 0.25x + 40 dollars for x minutes. An hour is 60 minutes. Pat says B's rate is much lower. Find his mistake. Word problem
- Celsius to Fahrenheit is linear. Water freezes at 0°C, which is 32°F. It boils at 100°C, which is 212°F. At what temperature do both scales show the same number? Word problem
- A piecewise function uses a different rule on each part. A data plan costs $30 for up to 5 GB. Each GB after 5 costs $8. Write the cost for x GB when x is more than 5. What do 12 GB cost? Word problem Above grade level
Answers
Tier 1 Foundations
- y = 150x + 1,200. 150 is liters added per minute. 1,200 is liters at the start.. The rate is the slope, m = 150. The starting value is the y-intercept, b = 1200.
- 2.75 is $2.75 per mile. 4 is the $4 at the start, before any miles.. The rate is the slope, m = 2.75. The starting value is the y-intercept, b = 4.
- y = 2x + 3. The slope is m = 2. Then b = y − mx = 3, the value of y when x = 0.
- y = 18x + 25. Ken swapped the numbers. The monthly rate goes with x, and the fee is the start: y = 18x + 25. The rate is the slope, m = 18. The starting value is the y-intercept, b = 25.
- After 16 hours. Solve 96 − 6x = 0. Check x = 16: left side 0, right side 0.
- y = 3x + 8. Plot (0, 8), (1, 11) and (2, 14).. y = 3x + 8: start at the y-intercept (0, 8). The slope is 3, so go up 3 and right 1 each time: (1, 11), then (2, 14).
Tier 2 Intermediate
- y = 15x + 120. The rate is the slope, m = 15. The starting value is the y-intercept, b = 120.
- After 10.5 minutes. Solve 18 + 4x = 60. Check x = 10.5: left side 60, right side 60.
- y = −0.08x + 48. The road drops 0.08 m for each meter. It starts 48 m high.. The slope is m = −0.08. Then b = y − mx = 48, the value of y when x = 0.
- It lands at 25 seconds. The model stops there.. The balloon lands at 25 seconds, when y = 0. The model doesn't work after that, so it can't be at −60 m. Solve 300 − 12x = 0. Check x = 25: left side 0, right side 0.
- y = −150x + 2,500. She drops 150 m an hour. She started at 2,500 m.. The slope is m = −150. Then b = y − mx = 2500, the value of y when x = 0.
- (0, 200), (1, 175), (2, 150). The line meets the x-axis at (8, 0), when the jar is empty.. y = −25x + 200: start at the y-intercept (0, 200). The slope is −25, so go down 25 and right 1 each time: (1, 175), then (2, 150).
Tier 3 Advanced
- y = 300x + 8,400. In 2030, x = 12, so 12,000 people.. The slope is m = 300. Then b = y − mx = 8400, the value of y when x = 0.
- y = −90x + 50 (1.5 cm a minute is 90 cm an hour). Work inside the parentheses first. Then multiply and divide. Then add and subtract. So 6/4 × 60 = 90.
- x = 0 is outside the model's range.. The model only works from 20 to 80 guests. x = 0 is outside that range, so $200 isn't a real price. The model only works for 20 to 80 guests. It says nothing about 0 guests.
- y = 1.5x + 9. 9 cm is its length with no weight. No: 0 cm would need −6 kg.. The slope is m = 1.5. Then b = y − mx = 9, the value of y when x = 0.
- y = −1.5x + 12. 12 is the soccer balls with no basketballs. Every 2 basketballs mean 3 fewer soccer balls.. Undo the formula one step at a time, in reverse order, until y is alone: y = 12 − 3x ÷ 2.
Tier 4 Challenge
- y = 12x − 600. 50 cakes. The −600 is the rent, owed before any sales.. Each cake makes 20 − 8 = 12 dollars. With no cakes, the bakery is down $600. So b = −600. Check x = 50: left side 0, right side 0.
- (10, 40) should be (10, 41). The first four points all fit y = 4x + 1. So the last one is the odd one out.
- The rates are the same: $0.25 a minute is $15 an hour.. Pat compared dollars per minute with dollars per hour. 0.25 × 60 = 15, so both charge $15 an hour. Change both rates to the same unit before you compare.
- −40 degrees. The model is F = 1.8C + 32. Set F = C: C = 1.8C + 32, so −0.8C = 32 and C = −40. Check x = −40: left side −40, right side −40.
- y = 30 + 8(x − 5), which is y = 8x − 10. 12 GB cost $86.. Work inside the parentheses first. Then multiply and divide. Then add and subtract. So 30 + 8 × (12 − 5) = 86.
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