Skills Beta

Lines: Slope and Intercept

A line y = a + bx has slope b , the change in y for each one-unit increase in x (rise over run), and y-intercept a , the value of y when x = 0 .

Practice 3: Analyze Data

Part 1 · Hook

Why this matters

A bike rental charges a 12 dollar fee plus 3 dollars an hour. Its cost is the line

y = 12 + 3x

. The 3 says how fast cost grows with time, and the 12 says where it starts. Unit 5 fits lines like this to data and asks you to interpret them in context, so the slope and intercept need to feel familiar now.

Part 2 · Prerequisite check

Quick check before you start

1. What is $12 + 3(5)$ ?

Show the answer

Multiply first: 15, then add 12.

Correct: 27:
75:
20:
15:

2. Evaluate $\frac{19 - 7}{6 - 2}$ .

3
12
4
0.33

Show the answer

12/4 = 3.

Correct: 3:
12:
4:
0.33:

Part 3 · See it

See it first

The line y = 12 + 3x for the cost in dollars of renting a bike for x hours, on axes from 0 to 8 hours and 0 to 40 dollars. The line starts at the intercept, 12 dollars at 0 hours. A step of 1 hour to the right (the run) goes with a rise of 3 dollars: the slope is 3 dollars per hour. — The line $y = 12 + 3x$ : it starts at 12 dollars and rises 3 dollars for each hour. LevlPrep original diagram.

Part 4 · Step by step

How it works, step by step

A line can be written $y = a + bx$ $a$ is the y-intercept and $b$ is the slope
Move one unit to the right on xy changes by $b$ : up if $b > 0$ , down if $b < 0$
Set $x = 0$ y equals $a$ , which has meaning only if $x = 0$ makes sense in context
Know two points, or one point and the slopeslope = rise over run, then solve for $a$ from one point

Part 5 · Key ideas

Key ideas

A line is $y = a + bx$ . The slope $b$ is the change in y for each one-unit increase in x; its units are y units per x unit.
Slope from two points: $b$ $= \frac{y_2 - y_1}{x_2 - x_1}$ , rise over run, subtracting in the same order on top and bottom.
The y-intercept $a$ is the value of y when $x = 0$ . Interpret it only when $x = 0$ makes sense.
In context: "For each additional hour, the cost increases by 3 dollars."

Part 6 · Misconception

A common mistake

The wrong idea: The line through $(3, 20)$ with slope 4 is $y = 20 + 4x$ .

What actually happens: 20 is the value of y at $x = 3$ , not at $x = 0$ . Solve $20 = a + 4(3)$ to get $a = 8$ : $y = 8 + 4x$ .

Part 7 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. A line passes through the points $(2, 7)$ and $(6, 19)$ . What is its slope?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

Slope = $\frac{\text{rise}}{\text{run}}$ $= \frac{19 - 7}{6 - 2}$ $= \frac{12}{4}$ $= 3$ .

Answer: 3

2. For the same line through $(2, 7)$ and $(6, 19)$ , what is the y-intercept?

Type a number. A table answer and a calculator answer are both accepted.

Show the answer

With slope 3, $y = a + 3x$ . Using $(2, 7)$ : $7 = a + 3(2)$ , so $a = 1$ .

Answer: 1

3. A bike rental costs $y = 12 + 3x$ dollars for $x$ hours. What does the slope 3 tell you?

The rental costs 3 dollars before any time is used
The cost goes up 3 dollars for each additional hour
The bike can be rented for at most 3 hours
The total cost of the rental is 3 times the hours

Show the answer

Slope = change in y per one-unit increase in x: 3 dollars more for each extra hour.

The rental costs 3 dollars before any time is used: That would be the intercept, which here is 12 dollars.
Correct: The cost goes up 3 dollars for each additional hour: Right: the slope is the change in y for each one-unit increase in x.
The bike can be rented for at most 3 hours: The slope is a rate, not a limit on x.
The total cost of the rental is 3 times the hours: That ignores the 12 dollar start; the cost is 12 plus 3 per hour.

4. For the rental cost $y = 12 + 3x$ , what does the y-intercept 12 tell you?

The cost goes up 12 dollars for each hour
The rental lasts 12 hours
The cost is 12 dollars per hour after the first hour
The cost is 12 dollars at 0 hours, a fixed fee

Show the answer

The y-intercept is the value of y when $x = 0$ : a 12 dollar fee before any hours.

The cost goes up 12 dollars for each hour: That would be a slope of 12; the slope here is 3.
The rental lasts 12 hours: The intercept is a value of y (dollars), not of x.
The cost is 12 dollars per hour after the first hour: After the start, each hour adds 3 dollars.
Correct: The cost is 12 dollars at 0 hours, a fixed fee: Right: the intercept is the value of y when x = 0.

5. A cooling drink’s temperature is $y = 80 - 2.5x$ degrees after $x$ minutes. What does the slope tell you?

The temperature rises 2.5 degrees for each minute
The drink starts at 2.5 degrees
The temperature falls 2.5 degrees for each minute
The drink is 80 degrees colder after each minute

Show the answer

Slope $-2.5$ : for each one-minute increase, the temperature decreases by 2.5 degrees.

The temperature rises 2.5 degrees for each minute: The minus sign means the temperature falls.
The drink starts at 2.5 degrees: The start, at $x = 0$ , is 80 degrees.
Correct: The temperature falls 2.5 degrees for each minute: Right: a negative slope means y goes down as x goes up.
The drink is 80 degrees colder after each minute: 80 is the starting temperature, not a rate.

6. A line for adults gives height in centimeters from shoe length in centimeters: $y = 95 + 2.8x$ . Why does the intercept 95 have no sensible meaning here?

A shoe length of 0 cm is impossible for an adult
No adult is 95 centimeters tall
The slope is positive, so the intercept is zero
Intercepts are given in different units from y

Show the answer

Interpret an intercept only when $x = 0$ makes sense in context; a shoe length of 0 does not.

Correct: A shoe length of 0 cm is impossible for an adult: Right: the intercept is y at $x = 0$ , far outside any real shoe length.
No adult is 95 centimeters tall: Some people are 95 cm tall; the problem is the x-value, 0.
The slope is positive, so the intercept is zero: A positive slope says nothing about the intercept’s value.
Intercepts are given in different units from y: The intercept is in y’s units, centimeters of height.

Part 8 · Summary

Summary

A line

y = a + bx

has slope

b

, the change in y for each one-unit increase in x (rise over run), and y-intercept

a

, the value of y when

x = 0

. Interpret both in context with units, and the intercept only when

x = 0

makes sense.

Part 9 · Up next

What comes next

Part 10 · Connections

Connections

Next: topic 5.1

Question set for this topic