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Construct a Function to Model a Linear Relationship

Students write a linear function to represent a relationship given two points, a table, or a graph by calculating the slope and y-intercept. They use the function to make predictions and solve problems within and beyond the data set.

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Construct a Function to Model a Linear Relationship

Grade 8 · Functions · question 1 of 3

0 of 20

20 rungs. A star at the top.

y = 7x + 3, y = 10, x =

How it is taught

0:00 of 4:09

A linear model has a constant rate and a starting amount.

Slope is the rate; the starting amount is the value at time 0.

Read it: Using a linear model both ways

A linear model y = 2x + 3 works both ways. Forward: at x = 4, multiply first, 2 × 4 = 8, then add 3, so y = 11. Backward: from y = 11, take the 3 away, 8, then divide by 2, so x = 4.

  1. To find y, multiply x by m first, then add b.
  2. To find x, undo the steps in reverse: take b away from y, then divide by m.
  3. Check by putting the answer back into the model.

The model says y is m times x, plus b, so finding y follows that order.

Finding x undoes it: the + b came last, so it comes off first, then the × m.

Both ways land on the same point, (4, 11), which is on the line.

A linear model y = mx + b gives y from x forward, and x from y backward by undoing in reverse.

Dividing before taking b away: 11 ÷ 2, then − 3.

Undo in reverse: the + 3 came last, so take it off first, 11 − 3 = 8, then 8 ÷ 2 = 4.

y = 12x + 6, x = 10, y = ___ — 126. Multiply first, 12 × 10 = 120, then add 6: 126.

Build y = 3x + 5 from points (2, 11) and (5, 20)

Maria saves money each week. After two weeks she has eleven dollars. After five weeks she has twenty. Can we find the rule?

Each measurement is a point: two comma eleven, and five comma twenty. Two points are enough to build a linear function.

Step one: find the slope, m. We use the rise over the run.

Plug the points in: twenty minus eleven on top, five minus two on the bottom.

That's nine over three, which equals three. So Maria saves three dollars every week.

Step two: find b, the starting amount. Start from y equals m x plus b.

Pick a point, say two comma eleven. Substitute: eleven equals three times two, plus b.

Three times two is six, so eleven equals six plus b.

Subtract six from both sides. b equals five — Maria started with five dollars.

Step three: put it together. Our function is y equals three x plus five.

Now let's use it. How much will she have after eight weeks?

Maria's SavingsWeek 2 → $11 savedWeek 5 → $20 savedWhat's the rule?Points: (2, 11) and (5, 20)Step 1 · Slope mm = (y₂ − y₁) / (x₂ − x₁)m = (20 − 11) / (5 − 2)m = 9 ÷ 3 = 3Step 2 · y-intercept by = m x + b11 = 3(2) + b11 = 6 + bb = 5Step 3 · Functiony = 3x + 5After 8 weeks?y = 3(8) + 5 = 29ANSWER$29after 8 weeks
Grade 8 · Maria's weekly savings

Substitute eight for x: three times eight, plus five.

Twenty-nine dollars. We modeled a real situation with a linear function — slope is the rate, b is the start!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
Climbing now
Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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