Linear relationships: slope-intercept form y = mx + b

Warm-up

Three quick questions from earlier lessons.

Discover

A taxi charges 3 dollars to start and 2 dollars per mile. Is the cost proportional to the distance?

No: it is linear, but starts at 3
Yes

Key idea

Every non-vertical line is $y = mx + b$ .

In

y = mx + b

,

m

is the slope (the rate of change) and

b

is the $y$ -intercept (the value at

x = 0

). When

b = 0

the relationship is proportional. Horizontal lines are

y = b

; vertical lines are

x = a

.

Worked example

Through $(0, 4)$ and $(3, 10)$ .

m = 2

(10 − 4) ÷ 3.

b = 4

Value at x = 0.

y = 2x + 4

Equation.

Your turn, guided

Your turn: the $y$ -intercept of $y = 7x - 5$ .

-5

b.

Practice ★

A gym charges 55 dollars to join and 19 dollars per month. Write the total cost $y$ after $x$ months.
A gym charges 77 dollars to join and 16 dollars per month. Write the total cost $y$ after $x$ months.

Which is the equation of the horizontal line through $(5, 1)$ ?
1. $y = 1$
2. $x = 5$
3. $y = 5$

Watch out!

Practice ★★

A line passes through $(0, 2)$ and $(1, 8)$ . Write its equation in the form $y = mx + b$ .
A line passes through $(0, 2)$ and $(2, 12)$ . Write its equation in the form $y = mx + b$ .

Boss problem. A candle is 21 cm tall when lit and burns 3 cm per hour. Let $y$ be its height after $x$ hours.

Exit ticket

Three questions to check you have got it.

A gym charges 55 dollars to join and 19 dollars per month. Write the total cost $y$ after $x$ months.
A gym charges 77 dollars to join and 16 dollars per month. Write the total cost $y$ after $x$ months.
A gym charges 25 dollars to join and 17 dollars per month. Write the total cost $y$ after $x$ months.

Summary card

Slope-intercept form

$y = mx + b$ : rate $m$ , start $b$ .

Watch out: the rate multiplies $x$ ; the starting amount stands alone.

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