Demonstration 1 of 4
Between the tick marks
Does the graph of d(t) = 12 - 3t include inputs that fall between the tick marks?
When every real t from 0 to 4 is allowed, the rule gives an output at each one, so the points join into a line. When only whole numbers are allowed, the graph is five separate points.
\[d(t)=12-3t\]
\[12-3(0.5)=10.5\]
t is the input, in time units. d(t) is the output, in depth units. The domain is the set of allowed inputs, and it decides whether the points join into a line.
Predict first. Switch to whole-number inputs only. What happens to the point at t = 0.5?
Choose an example
Constructed example: the chapter's rule d(t) = 12 - 3t, with a count-domain contrast.
Calculated values
- Domain
- every real t from 0 to 4
- Input t
- 0.5
- Output d(t)
- 12 - 3(0.5) = 10.5
d(0.5) = 12 - 3(0.5) = 12 - 1.5 = 10.5, so the point (0.5, 10.5) belongs to the graph. The domain allows every real t from 0 to 4, so the line between the tick marks is part of the story.
Use the idea
Before connecting dots on any graph, ask whether the inputs between them are allowed. Counts of tickets or people usually are not.
Where the conclusion applies
The chapter declares every real t from 0 through 4. The whole-number version is the same rule given a count domain, like the chapter's ticket model.
Check your understanding: In the interval domain, what output goes with t = 3.5?
Chapter 15 source: section "A line includes inputs between the tick marks". Demonstration C15-D01.
Demonstration 2 of 4
Slope from two points
From (-2, 5) to (2, -3), how fast does the output change, and does the order matter?
The dashed legs show the horizontal and vertical changes between the two points. Reversing both changes flips both signs, so the fraction is unchanged. Reversing one flips only one sign.
\[-8/4=-2\]
\[(5-(-3))/(-2-2)=8/(-4)=-2\]
Slope is vertical change divided by horizontal change. Both changes must go from the same start point to the same end point.
Predict first. Reverse only the denominator's order. Will the slope still be -2, or will its sign flip?
Choose an example
Constructed example: the chapter's points (-2, 5) and (2, -3), plus a vertical pair.
Calculated values
- Order
- first to second, both changes
- Vertical change
- -3 - 5 = -8
- Horizontal change
- 2 - (-2) = 4
- Slope
- -2
Vertical change -3 - 5 = -8; horizontal change 2 - (-2) = 4. Slope = -8/4 = -2. The output falls two units for each positive input unit, whichever point you start from.
Use the idea
Write both changes with the same start and end labels before dividing. A sign that disagrees with the picture of a falling line is a warning.
Where the conclusion applies
The axes increase to the right and upward. If both points share the same horizontal value, the line is vertical and its slope is undefined, not zero.
Check your understanding: What is the slope from (0, 1) to (2, -3)?
Chapter 15 source: section "Worked example: a negative slope from two coordinates". Demonstration C15-D02.
Demonstration 3 of 4
Intercepts and the domain
The line y = 2x + 3 crosses the x-axis at (-3/2, 0). Does that point mean anything in the plan?
Both intercepts belong to the full line. The shaded band is the plan's domain; only the solid part of the line describes the plan.
\[0 = 2x + 3\]
\[x = -3/2\]
An intercept is where a graph meets an axis. Set x = 0 to find the y-intercept and y = 0 to find the x-intercept. The plan domain is 0 ≤ x ≤ 6.
Predict first. Switch to rule B, y = x + 7. Is its x-intercept inside the plan domain?
Choose an example
Constructed example: the chapter's example plan rules A and B.
Calculated values
- Rule
- y = 2x + 3
- y-intercept
- x = 0 gives y = 3: (0, 3)
- x-intercept
- 0 = 2x + 3 gives x = -3/2: (-3/2, 0)
- x-intercept in the model?
- no, its input is negative
For y = 2x + 3: setting x = 0 gives y = 2(0) + 3 = 3. Setting y = 0: 0 = 2x + 3, so -3 = 2x and x = -3/2. The plan allows only 0 ≤ x ≤ 6, so (-3/2, 0) is an intercept of the line but has no meaning inside the plan.
Use the idea
When a model is used only for some inputs, check whether a calculated intercept lies among them before giving it a meaning.
Where the conclusion applies
The plans are invented. The algebra holds for every real x; the plan domain is a choice about which inputs the story covers.
Check your understanding: For y = x + 7, what is the y-intercept, and is it inside the plan domain?
Chapter 15 source: section "Intercepts need a domain". Demonstration C15-D03.
Demonstration 4 of 4
Where the lines cross
What does it mean that A and B cross at (4, 11)?
At each input, the two dots are the two outputs. They meet only at x = 4, where both rules give 11. Left of 4, B is higher; right of 4, A is higher.
\[A(x) = B(x)\]
\[2x + 3 = x + 7\]
A(x) = 2x + 3 and B(x) = x + 7 are the two example rules, for inputs from 0 through 6. The dotted line marks the chosen input.
Predict first. At x = 6, which rule gives the larger output, and by how much?
Choose an example
Constructed example: the chapter's rules A(x) = 2x + 3 and B(x) = x + 7 and their table.
Calculated values
- Input x
- 4
- A(x)
- 2(4) + 3 = 11
- B(x)
- 4 + 7 = 11
- Larger output
- tie: equal outputs
At x = 4: A(4) = 2(4) + 3 = 11 and B(4) = 4 + 7 = 11. The outputs are equal, so (4, 11) is the intersection. Solving 2x + 3 = x + 7 gives the same x = 4.
Use the idea
To check a crossing you read from a graph, put its input into both rules. Equal outputs confirm it; the graph alone is only an estimate.
Where the conclusion applies
The rules are invented and say nothing about which output is better. Solving 2x + 3 = x + 7 accounts for every real input, not only the four shown.
Check your understanding: At x = 2, what are A(2) and B(2)?
Chapter 15 source: section "An intersection means equal output". Demonstration C15-D04.