UNIT 1What a function isLesson 2 of 21 Relations and Functions2 Comparing Functions in Different Forms
Comparing Functions in Different Forms
Two functions can be shown in different ways. Find each one's rate and start, then compare.
What you'll be able to do
- Find the rate of change from a table, a graph, an equation or words.
- Find the starting value, even when the table skips x = 0.
- Tell a greater rate from a faster change.
- Plot two functions on one grid to compare them.
The big idea
Two numbers tell the story. A linear function has a rate of change and a starting value. To compare two functions, find both numbers for each.
An example. Function A has the table (0, 5), (2, 9), (4, 13). Function B is y = 3x + 2.
A's numbers. y goes up 4 when x goes up 2. So the rate is 4 ÷ 2 = 2. At x = 0, y = 5. So the start is 5.
B's numbers. In y = 3x + 2, the rate is 3 and the start is 2.
Compare. B grows faster, because 3 is more than 2. A starts higher, because 5 is more than 2.
No x = 0 in the table? Find the rate first. Then step back to x = 0.
Greater or faster? A rate of −6 is less than 4. But −6 changes faster, because 6 is more than 4. Faster means a bigger size. Ignore the sign.
Watch the units. 3 km in 40 minutes is not 3 km an hour. Change both to the same unit first.
Words you need
- Rate of change
- How much y changes when x goes up by 1.
- Starting value
- The value of y when x = 0. It is also called the initial value.
- Greater rate
- The bigger number. 4 is greater than −6.
- Changes faster
- The rate with the bigger size. −6 changes faster than 4.
Where to find the two numbers
| Shown as | Rate of change | Starting value |
|---|---|---|
| An equation y = mx + b | m | b |
| A table | Change in y ÷ change in x | y when x = 0 |
| A graph | Rise ÷ run | Where it crosses the y-axis |
| Words | 'per', 'each', 'a' | 'starts with', 'a fee of' |
Worked examples
Example 1
Function A has the table (0, 5), (2, 9), (4, 13). Function B is y = 3x + 2. Which has the greater rate of change?
- A goes up 4 when x goes up 2. Its rate is 4 ÷ 2 = 2.
- B's rate is the 3 in y = 3x + 2.
- 3 is more than 2, so B has the greater rate. A starts higher, at 5.
Answer: B
Example 2
Function A has the table (2, 11), (5, 20). Function B starts at 8 and goes up 2 each step. Which starts higher?
- A's rate is (20 − 11) ÷ (5 − 2) = 3.
- Step back 2 from (2, 11): 11 − 2 × 3 = 5. A starts at 5.
- B starts at 8. So B starts higher.
Answer: B
Example 3
Graph y = 2x − 1 and y = −x + 5 on one grid. For which x is the first one higher?
- Line 1: plot (0, −1). Go up 2 and right 1 to (1, 1), then (2, 3).
- Line 2: plot (0, 5). Go down 1 and right 1 to (1, 4), then (2, 3).
- They cross at (2, 3). To the right of x = 2, Line 1 is higher.
Answer: (0, −1), (1, 1), (2, 3). (0, 5), (1, 4), (2, 3). They cross at (2, 3)
Common mistakes
Using the first row as the start. The start is y when x = 0. If the table starts at x = 2, step back to 0.
Dividing y by x in one row. That works only for lines through (0, 0). Use the change in y ÷ the change in x.
Calling −6 a greater rate than 4. −6 is less than 4. It changes faster, because its size is 6.
Thinking lines that cross have the same rate. Crossing means the same y at one x. The rates can be very different.
Practice: 22 questions
13 of 22 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Function A has the table (0, 3), (2, 11), (4, 19). Function B is y = 3x + 5. Which has the greater rate of change?
- Mia starts with $40 and saves $15 a week. Jay's savings are y = 20x + 25 dollars after x weeks. Who started with more? Word problem
- The graph shows Line A, through (0, 4) and (3, −2). Function B is y = −x + 1. Which has the greater rate of change?
- A bus pass costs $5 for 1 ride, $7 for 2 rides and $9 for 3 rides. A card costs $4 plus $2.50 a ride. Zoe says the pass starts at $5. Find her mistake. Which starts higher? Word problem
- Candle A is 20 cm tall and burns down 3 cm every 2 hours. Candle B's height is y = 20 − 1.2x cm after x hours. Which candle burns faster? Word problem
- You graph y = −2x + 6 and y = (1/2)x + 1 on one grid. Which three points do you plot for each, starting at the y-intercept? Which line goes down?
Tier 2 Intermediate One more step to think through
- Plan A costs $58 for 2 months, $97 for 5 months and $149 for 9 months. Plan B costs $35 to start plus $12 a month. How much more per month does Plan A cost? How much more does Plan B cost to start? Word problem
- Function f is y = −6x + 2. The graph shows Line g, through (0, 1) and (2, 9). Which has the greater rate of change? Which changes faster?
- Tank A holds 500 L and drains 20 L a minute. Tank B holds 380 L at 0 minutes, 330 L at 5 and 280 L at 10. Which tank holds more at 10 minutes? At 15 minutes? Word problem
- Line A is y = 2x + 1. Line B is y = −x + 10. They cross at (3, 7). Nia says they have the same rate of change because they meet. Find her mistake.
- Rosa walks 3 km every 40 minutes. Dev walks d = 4.2t km in t hours. An hour is 60 minutes. Who walks faster? Word problem
- Function A has the table (−2, 7), (0, 3), (2, −1). Function B is y = (3/4)x − 2. Plot each from its y-intercept. Which is greater at x = 4?
Tier 3 Advanced Several steps - find the hidden structure
- Pia has $160 and spends $12 a week. Raj has $40 at week 2, $76 at week 5 and $112 at week 8. In which week do they have the same amount? Who has more after that? Word problem
- Plan A costs y = 8x + 30. Plan B costs $47 for 1 month, $61 for 3 and $82 for 6. Plan C costs $25 to start plus $9 a month. Rank the plans by monthly cost, least first. Word problem
- The graph shows Line A, through (−1, 8) and (3, 0). Function B is y = kx + 4. For which k do they have the same rate? Then which has the greater initial value?
- On a savings graph, each square up is $10. Each square across is 2 weeks. Line A rises 3 squares for each square across. Omar saves $25 a week. He says Line A saves faster, because 3 squares is $30. Find his mistake. Word problem
- Car A starts with 50 L of fuel and uses 6 L every 100 km. Car B has 42 L at 100 km and 30 L at 300 km. Which car uses fuel faster? Which runs out first? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- Two taxis each charge a start fee plus a rate per km. A 4 km ride costs $14 with both. A 10 km ride costs $29 with both. Kemi says one taxi has a lower rate. Can she be right? Word problem
- In a 100 m race, Ben starts 30 m ahead and runs 5 m each second. Ama starts at the line. She is at 14 m after 2 seconds and 28 m after 4 seconds. Does Ama catch Ben before the finish? Word problem
- Function A has the table (0, 10), (3, 8), (6, 6). Function B goes up 1 each time x goes up 3. B passes through (3, 3.5). For which x is B above A?
- Plan A costs $60 to join plus $12 a month. Plan B costs $90 to join. Plan B must cost less than Plan A at 12 months. Which monthly rates work for Plan B? Word problem
- The average rate of change between two points is rise ÷ run. Function A is y = x². Function B is y = 5x − 3. From x = 1 to x = 5, which has the greater average rate of change? From x = 0 to x = 2? Above grade level
Answers
Tier 1 Foundations
- A. A's rate of change is 4. B's is 3.
- Mia. Mia's initial value is 40. Jay's is 25.
- B. Line A's rate of change is −2. B's is −1.
- The card starts higher. The pass starts at $3.. $5 is the cost of 1 ride. The start is the cost at 0 rides. Step back $2: the pass starts at $3. The pass's initial value is 3. The card's is 4.
- Candle A. Candle A's rate is −1.5 and Candle B's is −1.2. Ignore the signs: 1.5 against 1.2. The bigger size changes faster.
- y = −2x + 6: (0, 6), (1, 4), (2, 2). y = (1/2)x + 1: (0, 1), (2, 2), (4, 3). The first line goes down.. y = −2x + 6: start at the y-intercept (0, 6). The slope is −2, so go down 2 and right 1 each time: (1, 4), then (2, 2). y = (1/2)x + 1: start at the y-intercept (0, 1). The slope is 1/2, so go up 1 and right 2 each time: (2, 2), then (4, 3).
Tier 2 Intermediate
- Plan A costs $1 more per month ($13, against $12). Plan B costs $3 more to start ($35, against $32).. Plan A's rate is 39 ÷ 3 = 13. Step back 2 months: 58 − 26 = 32. So A starts at $32.
- g has the greater rate (4 is greater than −6). f changes faster (6 is more than 4).. f's rate of change is −6. g's is 4.
- Tank A at 10 minutes (300 L, against 280 L). Tank B at 15 minutes (230 L, against 200 L).. Tank B drains (380 − 330) ÷ 5 = 10 L a minute. At 10 minutes: A has 300 L and B has 280 L. At 15 minutes: A has 200 L and B has 230 L.
- They cross at (3, 7), but A's rate is 2 and B's is −1.. Meeting means the same y at x = 3. It doesn't mean the same rate. A's rate is 2 and B's is −1. (3, 7) makes both equations true: check it in y = 2x + 1 and y = −x + 10.
- Dev < Rosa, so Rosa is the fastest. Speeds in km an hour: Dev 4.2, Rosa 4.5.
- A: (0, 3), (1, 1), (2, −1). B: (0, −2), (4, 1), (8, 4). At x = 4, B is greater (1, against −5).. y = −2x + 3: start at the y-intercept (0, 3). The slope is −2, so go down 2 and right 1 each time: (1, 1), then (2, −1). y = (3/4)x − 2: start at the y-intercept (0, −2). The slope is 3/4, so go up 3 and right 4 each time: (4, 1), then (8, 4).
Tier 3 Advanced
- Week 6, when both have $88. After that, Raj has more.. (6, 88) makes both equations true: check it in y = 160 − 12x and y = 40 + (76 − 40)/(5 − 2) · (x − 2).
- B ($7), A ($8), C ($9). By start fee it is C ($25), A ($30), B ($40).. Monthly costs: A 8, B 7, C 9.
- k = −2. A has the greater initial value (6, against 4).. The slope is m = −2. Then b = y − mx = 6, the value of y when x = 0.
- Line A < Omar, so Omar is the faster saver. Each square across is 2 weeks. So Line A saves $30 every 2 weeks, which is $15 a week. Omar saves faster. Savings each week: Line A 15, Omar 25.
- They use fuel at the same rate, 6 L per 100 km. Car B runs out first, because it starts with less (48 L).. Car A's rate of change is −0.06. Car B's is −0.06.
Tier 4 Challenge
- No. Both taxis are y = 2.5x + 4.. Two points fix one line. Both taxis pass through (4, 14) and (10, 29), so they are the same function.
- No. She would catch him at 15 seconds, at 105 m. That is past the finish.. Ama runs 7 m a second. 7x = 30 + 5x gives x = 15 and 105 m. The race ends at 100 m, so she never catches him.
- For x greater than 7.5. They meet at (7.5, 5).. (7.5, 5) makes both equations true: check it in y = 10 − (2/3) · x and y = (1/3) · x + 2.5.
- Any rate below $9.50 a month. Plan A costs $60 + $144 = $204 at 12 months. Plan B costs 90 + 12r. That is less than 204 when r is below 9.50.
- From 1 to 5, A (6, against 5). From 0 to 2, B (5, against 2).. For A from 1 to 5: (25 − 1) ÷ 4 = 6. From 0 to 2: (4 − 0) ÷ 2 = 2. B's rate is always 5.
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