UNIT 2Linear functionsLesson 1 of 31 Linear and Nonlinear Functions2 Rate of Change and Initial Value3 Describing Graphs: Increasing and Decreasing
Linear and Nonlinear Functions
A linear function changes by the same amount every step. Its graph is a straight line.
What you'll be able to do
- Test a table: check the rate between every pair of neighbors.
- Test an equation: can it be written as y = mx + b?
- Plot a table and see if the points line up.
- Tell a steady change from one that speeds up.
The big idea
Linear means steady. A linear function changes by the same amount each step. Its graph is a straight line.
Test a table. Find the rate between each pair of neighbors. That is the change in y ÷ the change in x. If every rate matches, it is linear.
An example. x: 0, 1, 2, 3 and y: 1, 4, 9, 16. The changes are 3, 5, 7. They don't match, so it is not linear.
Uneven steps. Sometimes x jumps by 2, then by 1. Divide each change in y by its change in x. Don't just compare the changes in y.
Test an equation. Solve for y. If you get y = mx + b, it is linear. y = 2x − 3 is linear. y = x² + 2 and y = 6/x are not.
Simplify first. y = x(x + 3) − x² has squares. But it simplifies to y = 3x, which is linear.
Test a graph. A straight line is linear. A curve or a bend is not.
Test a story. Adding the same amount each time is linear. Doubling each time is not.
Words you need
- Linear function
- A function whose graph is a straight line. It changes by the same amount each step.
- Nonlinear function
- A function whose graph is not a straight line. Its rate of change is not the same.
- Rate of change
- The change in y divided by the change in x.
- Neighbors
- Two rows next to each other in a table.
- Simplify
- Multiply out and combine terms, so the rule is as short as it can be.
- Product
- The answer when you multiply. The product of 3 and 4 is 12.
Linear or not?
| Shown as | Linear | Not linear |
|---|---|---|
| Equation | y = 4 − 3x, y = x/5, 2x + y = 5 | y = x² + 1, y = 5/x, xy = 12 |
| Table | Same rate between every pair | The rate changes |
| Graph | A straight line | A curve or a bend |
| Story | Adds the same amount | Doubles, or adds more each time |
Worked examples
Example 1
Is the table x: 0, 1, 2, 3 and y: 1, 4, 9, 16 linear?
- Find each change in y: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7.
- The x-steps are all 1, but the changes are 3, 5, 7.
- They don't match. It is not linear.
Answer: Not linear
Example 2
Is 4x − 2y = 6 linear? Is y = x² + 2?
- Solve 4x − 2y = 6 for y: −2y = −4x + 6, so y = 2x − 3.
- That is y = mx + b, so it is linear.
- y = x² + 2 has an x². It is not linear.
Answer: Linear: 4x − 2y = 6
Example 3
Plot y = 2x − 1 and y = x² − 1 for x = 0, 1, 2 and 3. Which one lines up?
- y = 2x − 1 gives (0, −1), (1, 1), (2, 3), (3, 5). Each point is up 2 from the last.
- So those points line up.
- y = x² − 1 gives (0, −1), (1, 0), (2, 3), (3, 8). The rises are 1, 3, 5.
- The rises grow, so those points bend.
Answer: (0, −1), (1, 1), (2, 3), (3, 5). Then (0, −1), (1, 0), (2, 3), (3, 8)
Common mistakes
Checking only the first two rows. Check every pair of neighbors. One different rate means it is not linear.
Comparing changes in y when x jumps unevenly. Divide each change in y by its change in x.
Thinking linear must pass through (0, 0). y = 2x + 3 is linear. It crosses at (0, 3).
Calling x/5 and 5/x the same. x/5 is linear. 5/x has x on the bottom, so it is not.
Practice: 22 questions
12 of 22 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Which of these are linear: 2x + y = 5, y = x² + 1, y = x/5, y = 5/x?
- A diver is 2 m deep at 0 seconds, 5 m at 1 second, 8 m at 2 and 11 m at 3. Is her depth a linear function of time? Word problem
- Gil says y = 2x + 3 is not linear because its graph misses (0, 0). Find his mistake.
- A video has 50 views on day 1, 100 on day 2, 200 on day 3 and 400 on day 4. The views double each day. Is the number of views a linear function of the day? Word problem
- A lot charges $10 for a stay of 1, 2, 3 or 4 hours. Is the cost a linear function of the hours for those stays? Word problem
- Make a table for y = x² − 2 at x = −2, −1, 0, 1 and 2. Which points do you plot? Do they lie on one line?
Tier 2 Intermediate One more step to think through
- A plant is 4 cm tall on day 0. It is 10 cm on day 2, 13 cm on day 3 and 25 cm on day 7. Is its height a linear function of the day? Word problem
- Is y = x(x + 3) − x² a linear function?
- A taxi costs $5 for 1 mile, $7 for 2 miles, $9 for 3 miles and $14 for 5 miles. Sami checks the first two rows. He sees +$2 and says it is linear. Find his mistake. Word problem
- A cylinder's volume is V = πr²h. Keep r = 2 cm. Is V a linear function of the height h? Now keep h = 5 cm. Is V a linear function of r? Word problem
- Graphs A, B and C are shown. A passes through (−2, 7), (0, 3) and (1, 1). B passes through (0, 0), (1, 1) and (2, 4). C joins (0, 0), (2, 2) and (4, 0). Which show a linear function?
- A table has x: −1, 0, 2, 3 and y: 5, 3, −1, −3. Plot the points. If it is linear, which point comes next at x = 4?
Tier 3 Advanced Several steps - find the hidden structure
- A data plan costs $7 for 1 GB, $13 for 3 GB, $k for 4 GB and $28 for 8 GB. The cost is linear. Find k. Word problem
- Is y = 2(3x − 1) − (6x − 5) linear? Describe its graph.
- Gym B costs $20 the first month. Each month costs $5 more than the month before. Is the total paid a linear function of the months? Word problem
- A ball's height is 0 m at 0 s, 3.75 m at 0.5 s and 5 m at 1 s. Lia says it is linear up to 1 s because the height rises both times. Find her mistake. Word problem
- Which of these is not linear: y = 0.5(4 − 2x), 3x + 2y = 7, y = x(x − 1) + 5 − x², xy = 12?
Tier 4 Challenge Stretch problems, some above grade level
- Every rectangle in a set has an area of 24 cm². Is the perimeter a linear function of the width? Check widths 1, 2 and 3 cm. Word problem
- A car rental costs $40 a day, but every 7th day is free. Is the cost a linear function of the days? Word problem
- Jin says y = 3x + 1 and y = 2x − 4 are both linear. So he says y = (3x + 1)(2x − 4) is linear too. Find his mistake.
- For which a is y = x(ax − 5) − 2(3x² − x) + 7 a linear function?
- Plan A pays $100 a day. Plan B pays 1 cent on day 1. Then it doubles each day: 1, 2, 4, 8 cents. Doubles means it is multiplied by 2. On which day does Plan B first pay more than Plan A for that day? Word problem Above grade level
Answers
Tier 1 Foundations
- Linear: 2x + y = 5, x/5. 2x + y = 5: y = 5 − 2x, linear. x² + 1: not linear. x/5: y = x/5, linear. 5/x: not linear.
- Yes. She goes 3 m deeper every second.. Every step has the same rate of change, 3, so it's linear.
- Linear. Linear means y = mx + b. The graph is a straight line. It just crosses the y-axis at (0, 3). y = 2x + 3 can be written as y = mx + b, so it's linear.
- No. The changes are 50, 100 and 200.. The rate of change between neighboring points isn't constant, so it's not linear.
- Yes. The cost is y = 10. The rate of change is 0.. Every step has the same rate of change, 0, so it's linear.
- (−2, 2), (−1, −1), (0, −2), (1, −1), (2, 2). No. They make a U shape.. Put each x into y = x² − 2: (−2, 2), (−1, −1), (0, −2), (1, −1), (2, 2).
Tier 2 Intermediate
- Yes. Every step gives 3 cm a day: 6 ÷ 2, 3 ÷ 1 and 12 ÷ 4.. Every step has the same rate of change, 3, so it's linear.
- Yes. It simplifies to y = 3x.. y = x · (x + 3) − x² can be written as y = mx + b, so it's linear.
- Not linear. Sami stopped too soon. From 3 to 5 miles, the rate is 5 ÷ 2 = $2.50 a mile, not $2. The rate of change between neighboring points isn't constant, so it's not linear.
- Yes: V = 4πh. No: V = 5πr² has r squared.. 2²x: y = 4x, linear. 5x²: not linear.
- Only Graph A. Graph A: one straight line. Graph B: the rate changes, so it bends. Graph C: the rate changes, so it bends.
- It is linear: the rate is −2 every step. The next point is (4, −5).. y = −2x + 3: each point makes the equation true, so it is on the line.
Tier 3 Advanced
- k = 16 (the rate is $3 per GB). Work inside the parentheses first. Then multiply and divide. Then add and subtract. So 7 + (13 − 7) ÷ (3 − 1) × (4 − 1) = 16.
- Yes. It simplifies to y = 3, a flat line.. Multiply each term inside the parentheses. Then combine like terms. So 2 · (3x − 1) − (6x − 5) = 3.
- No. The totals grow by more each month: 25, then 30, then 35.. The months cost $20, $25, $30. The totals are $20, $45, $75. They grow by 25, then 30, so the rate changes.
- Not linear. Rising isn't enough. The rates are 3.75 ÷ 0.5 = 7.5 and 1.25 ÷ 0.5 = 2.5 m a second. They differ. The rate of change between neighboring points isn't constant, so it's not linear.
- xy = 12. It means y = 12/x. The other three simplify to y = mx + b.. 0.5(4 − 2x): y = 2 − x, linear. 3x + 2y = 7: y = 7/2 − 3x/2, linear. x(x − 1) + 5 − x²: y = 5 − x, linear. xy = 12: not linear.
Tier 4 Challenge
- No. The perimeters are 50, 28 and 22 cm. They change by −22, then −6.. A fixed area means the length is 24 ÷ width. Dividing by x is not linear.
- No. Day 6 and day 7 both cost $240.. A linear rate never changes. The free day changes it from 40 to 0.
- The product is not linear. The sum, 5x − 3, is linear.. The product has 3x · 2x = 6x². So it is not linear. Adding two linear functions keeps it linear. Multiplying does not. Multiplying two x terms makes an x² term.
- a = 6. Then y = −3x + 7.. Multiply out: ax² − 5x − 6x² + 2x + 7. The x² terms cancel only when a = 6.
- Day 15 (16,384 cents is $163.84). Day n pays 2ⁿ⁻¹ cents. Day 14 pays 8,192 cents, less than 10,000. Day 15 pays 16,384 cents.
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