Skills

Slopes and Curve Shapes

Slope measures how steeply a line rises or falls: the rise (vertical change) divided by the run (horizontal change).

Practice 4: Graphing and Visuals

Part 1 · Hook

Why this matters

A delivery van burns a lot of fuel per mile crawling through town at 20 miles an hour, less at 50 on an open road, and more again at 70 as wind resistance climbs. Plot fuel against speed and you get a U: the curve falls, bottoms out, then rises. Reading which way a curve slopes, and how steeply, is how you read every graph in economics.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. With quantity across and price up, the ordered pair (40, 5) means

  1. a quantity of 40 at a price of 5
  2. a price of 40 at a quantity of 5
  3. a total of 45
  4. a quantity of 5 at a price of 40
Show the answer

The first number is the horizontal value, the second the vertical value.

  • Correct: a quantity of 40 at a price of 5:
  • a price of 40 at a quantity of 5:
  • a total of 45:
  • a quantity of 5 at a price of 40:

2. Points that fall as you move to the right show

  1. an inverse relationship
  2. a direct relationship
  3. no relationship
  4. a mistake in the table
Show the answer

One variable rises while the other falls.

  • Correct: an inverse relationship:
  • a direct relationship:
  • no relationship:
  • a mistake in the table:

Part 4 · See it

See it first

Four small graphs: an upward line with its rise of 3 over a run of 4 marked (slope 0.75), a downward line (negative slope), a horizontal and a vertical line, and a U-shaped curve with its minimum point marked.
Four shapes. Upward: a rise of 3 over a run of 4 gives a slope of 0.75. Downward: a negative slope. A horizontal line has a slope of 0. A U-shaped curve falls, reaches its minimum point, then rises. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. You move between two points on a linethe vertical change is the rise and the horizontal change is the run
  2. You divide the rise by the runyou get the slope: how much the vertical variable changes for each one-unit change in the horizontal one
  3. The rise and the run have the same signthe slope is positive and the line rises to the right
  4. The rise and the run have opposite signsthe slope is negative and the line falls to the right

Part 6 · Key ideas

Key ideas

  • Slope = rise over run = (change in the vertical value) divided by (change in the horizontal value). Subtract in the same order on top and bottom.
  • Positive slope: the line rises to the right (a direct relationship). Negative slope: it falls to the right (an inverse relationship). A horizontal line has a slope of 0.
  • A straight line has the same slope everywhere. A curve has a slope that changes from point to point.
  • A U-shaped curve falls, then rises. At its minimum point the slope is 0: the curve stops falling and starts rising.

Part 7 · Misconception

A common mistake

The wrong idea: Slope is the run divided by the rise: the horizontal change over the vertical change.

What actually happens: Slope is the rise over the run: the vertical change divided by the horizontal change. From (20, 8) to (60, 4) the slope is (4 - 8) / (60 - 20) = -0.1, not -10.

Part 8 · Check yourself

Check yourself

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

1. What is the slope of a line?

  1. The change in the horizontal value divided by the change in the vertical value
  2. The vertical value at the point where the line starts
  3. The change in the vertical value divided by the change in the horizontal value
  4. The sum of the two values at any point on the line
Show the answer

Slope is rise over run: vertical change on top, horizontal change on the bottom.

  • The change in the horizontal value divided by the change in the vertical value: That is run over rise, the slope turned upside down.
  • The vertical value at the point where the line starts: The starting height says where the line is, not how steep it is.
  • Correct: The change in the vertical value divided by the change in the horizontal value: Right: rise over run.
  • The sum of the two values at any point on the line: Adding the coordinates of a point does not measure steepness.

2. A straight line passes through (20, 8) and (60, 4). What is its slope?

Type a number. A four-function calculator is allowed, as on the exam.

Show the answer

Rise: 4−8=−44 - 8 = -4. Run: 60−20=4060 - 20 = 40. Slope =−440=−0.1= \frac{-4}{40} = -0.1, negative because the line falls to the right.

  • Answer: -0.1

3. What is the slope of a horizontal line?

  1. 0
  2. 1
  3. A negative number
  4. The height of the line
Show the answer

Along a horizontal line the vertical value never changes, so the rise is 0 and 0 divided by any run is 0.

  • Correct: 0: Right: no rise, so the slope is 0.
  • 1: A slope of 1 rises one unit for each unit across.
  • A negative number: A negative slope falls to the right; a horizontal line does not fall.
  • The height of the line: The height says where the line sits, not how steep it is.

4. Line P falls 3 units for every 1 unit across. Line Q falls 1 unit for every 3 units across. Which statement is correct?

  1. Both slopes are negative, and Q is steeper
  2. P has a positive slope and Q a negative one
  3. They have the same slope because both fall
  4. Both slopes are negative, and P is steeper
Show the answer

P has a slope of -3 and Q a slope of about -0.33. Both fall, and the one with the larger size (3) is steeper.

  • Both slopes are negative, and Q is steeper: Q falls only one unit for three across, so it is flatter.
  • P has a positive slope and Q a negative one: Both lines fall to the right, so both slopes are negative.
  • They have the same slope because both fall: Falling tells you the sign, not the size; -3 and -1/3 differ.
  • Correct: Both slopes are negative, and P is steeper: Right: -3 is steeper than -1/3.

Data

A delivery van's fuel use at different speeds

Gallons of fuel used per 100 miles at each steady speed (invented data for practice)
Speed (miles per hour)Fuel used (gallons per 100 miles)
206.0
304.6
403.8
503.4
603.6
704.4

5. At what speed does the van reach the minimum point of its fuel curve?

  1. 20 miles per hour
  2. 50 miles per hour
  3. 70 miles per hour
  4. 40 miles per hour
Show the answer

Fuel use falls to 3.4 gallons at 50 miles per hour, then rises again. The lowest value marks the minimum point.

  • 20 miles per hour: At 20 miles per hour the van uses the most fuel in the table.
  • Correct: 50 miles per hour: Right: 3.4 gallons is the lowest value.
  • 70 miles per hour: At 70 miles per hour fuel use is rising again.
  • 40 miles per hour: At 40 miles per hour fuel use is still falling.

6. Place point X on the downward-sloping curve D at a larger quantity than the point where D meets the upward-sloping curve S.

QuantityPriceDSQP

Work on the graph (it needs JavaScript), then check.

Show the answer

D slopes down, so moving right along it to a larger quantity takes you to a lower price: X goes down and to the right along D.

  • Key: see the explanation

Part 9 · Summary

Summary

Slope measures how steeply a line rises or falls: the rise (vertical change) divided by the run (horizontal change). A positive slope rises to the right, a negative slope falls to the right, and a horizontal line has a slope of 0. A straight line keeps one slope; a curve changes slope as you move along it. A U-shaped curve has a negative slope, then a slope of 0 at its minimum point, then a positive slope.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic