SHSAT: Linear functions: slope, intercepts and rates
Find slope from two points, write y = mx + b from a point or a story, and find intercepts and values.
Given two points or a rate and a starting amount, how do you write and use a linear function?
Short answer
Slope is change in y divided by change in x. Write y = mx + b, find b by substituting a known point, and set y = 0 to find where the line crosses the x-axis.
Know first: Subtracting negative numbers, Plotting points on a coordinate plane
Step 1 of 5
Slope is change in y over change in x
A linear function changes by the same amount every time x goes up by 1. That amount is the slope. Between any two points, slope = (change in y) ÷ (change in x). Subtract in the same order on top and bottom.
In y = mx + b, m is the slope and b is the y-intercept, the value of y when x = 0. In a story, the slope is the rate and b is the starting amount or fixed fee.
| You want | Do this |
|---|---|
| slope from two points | (y₂ − y₁) ÷ (x₂ − x₁) |
| y-intercept from slope and a point | b = y − mx, using the point's x and y |
| x-intercept | set y = 0 and solve for x |
| a parallel line | keep the same slope, then find the new b |
| a rule from a table | slope from any two rows, then b from one row; check a third row |
Grade 9 SHSAT forms cover math through Grade 8, which includes linear functions.
Step 2 of 5
Keep the subtraction order the same
Worked example: Slope from two points
A line passes through (−2, 5) and (3, −5). What is its slope?
- A−2
- B−1/2
- C1/2
- D2
- 1
Change in y
−5 − 5 = −10.
- 2
Change in x, same order
3 − (−2) = 5.
- 3
Divide
−10 ÷ 5 = −2. The line falls as x increases, so a negative slope makes sense.
- 4
See why the other choices are there
−1/2 divides change in x by change in y. 2 subtracts the y-values in one order and the x-values in the other. 1/2 makes both mistakes.
Answer
−2
A fixed fee from two data points
A plumber charges a fixed fee plus a constant hourly rate. 2 hours cost $170 and 5 hours cost $335.
Rate: (335 − 170) ÷ (5 − 2) = 165 ÷ 3 = $55 per hour
Fixed fee: 170 − 2 × 55 = 170 − 110 = $60
Check: 60 + 5 × 55 = 335
Find the wrong step
A line has slope 3/4 and passes through (8, 2). Find its y-intercept.
- 1
Write y = (3/4)x + b.
- 2
Substitute: 8 = (3/4)(2) + b, so b = 8 − 3/2 = 13/2.
What went wrong
The point (8, 2) has x = 8 and y = 2. The values were swapped.
- 3
The y-intercept is 13/2.
The fix
Substitute x = 8 and y = 2: 2 = (3/4)(8) + b = 6 + b, so b = −4.
Step 3 of 5
Rates, starting amounts and intercepts
Check yourself · Question 1
What is the x-intercept of y = −4x + 10?
Answer: B
Set y = 0: 0 = −4x + 10, so 4x = 10 and x = 5/2. Check: −4(5/2) + 10 = 0.
Check yourself · Question 2
A pool holds 600 liters at 9:00 and 480 liters at 9:20. It drains at a constant rate. How many minutes after 9:20 will it hold 240 liters?
Answer: A
The rate is (600 − 480) ÷ 20 = 6 liters per minute. From 480 to 240 is 240 liters, which takes 240 ÷ 6 = 40 minutes.
Trap answer: 60
Measuring from the wrong starting time
- Why it looks right
- 60 minutes is correct from 9:00, the first time in the problem.
- Why it's wrong
- The question measures from 9:20. Twenty of those 60 minutes have already passed by then.
- The right answer
- 40 minutes, from 480 − 240 = 240 liters at 6 liters per minute.
Check yourself · Question 3
Which statement is true for y = −3x + 6?
Answer: B
At x = 2, y = −3(2) + 6 = 0, so (2, 0) is on the line. It's the x-intercept.
Common mistake
I subtract the y-values in one order and the x-values in the other.
- Why it's tempting
- Each subtraction feels natural on its own, especially when you pick the order that avoids negatives.
- Do this instead
- Label the points 1 and 2 before you start. Then compute y₂ − y₁ and x₂ − x₁, always in that order.
Step 4 of 5
What to remember
Remember
- Slope = change in y ÷ change in x, with the same subtraction order in both.
- In y = mx + b, m is the rate and b is the starting value; find b by substituting a point.
- Find an x-intercept by setting y = 0, and keep the slope when you write a parallel line.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, which point did you call the first point, and what were the change in y and the change in x?