6.4 - Interpret Initial Value and Rate of Change in Equations | MFM1P - Free Printable

Let’s solve this step by step.

We are given three menus, each with a cost formula:

- Menu 1: C = 10n + 100
- Menu 2: C = 14n + 100
- Menu 3: C = 18n + 100

Where:
- n = number of people
- C = total cost in dollars

We need to complete the tables for each menu. The “First Difference” is the change in cost when you go from one row to the next (since n increases by equal amounts, the first difference should be constant for linear equations like these).

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Menu 1: C = 10n + 100

n goes up by 25 each time: 25, 50, 75, 100, 125

Calculate C for each:

- n=25 → C = 10*25 + 100 = 250 + 100 = 350
- n=50 → C = 10*50 + 100 = 500 + 100 = 600
- n=75 → C = 10*75 + 100 = 750 + 100 = 850
- n=100 → C = 10*100 + 100 = 1000 + 100 = 1100
- n=125 → C = 10*125 + 100 = 1250 + 100 = 1350

Now find First Differences (change in C):

- 600 - 350 = 250
- 850 - 600 = 250
- 1100 - 850 = 250
- 1350 - 1100 = 250

So first difference is always 250. That makes sense because every time n increases by 25, and cost per person is $10, so 25 * 10 = 250.

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Menu 2: C = 14n + 100

n goes up by 50: 0, 50, 100, 150, 200

Calculate C:

- n=0 → C = 14*0 + 100 = 0 + 100 = 100
- n=50 → C = 14*50 + 100 = 700 + 100 = 800
- n=100 → C = 14*100 + 100 = 1400 + 100 = 1500
- n=150 → C = 14*150 + 100 = 2100 + 100 = 2200
- n=200 → C = 14*200 + 100 = 2800 + 100 = 2900

First Differences:

- 800 - 100 = 700
- 1500 - 800 = 700
- 2200 - 1500 = 700
- 2900 - 2200 = 700

Again, consistent: 50 people * $14 = $700 increase each time.

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Menu 3: C = 18n + 100

The problem says “n must go up by equal increments”. Looking at the other two, they chose increments that make sense for their context. For Menu 3, we can choose any increment — but since it’s not specified, let’s pick an increment that’s easy and matches the pattern. Let’s use n going up by 25, same as Menu 1, to keep things simple. (You could also choose 50 or 10 — but 25 is fine.)

So let’s set n: 0, 25, 50, 75, 100

Wait — actually, looking back at the table structure for Menu 3, it has 5 rows blank, just like the others. And the note says “*n goes up by ___” — so we get to choose! But to match the style, let’s pick n going up by 25 again, starting from 0? Or maybe start from 25? Actually, Menu 2 started at 0, Menu 1 started at 25. To be consistent with having 5 values, let’s do:

n: 0, 25, 50, 75, 100

But wait — if we start at 0, then:

- n=0 → C = 18*0 + 100 = 100
- n=25 → C = 18*25 + 100 = 450 + 100 = 550
- n=50 → C = 18*50 + 100 = 900 + 100 = 1000
- n=75 → C = 18*75 + 100 = 1350 + 100 = 1450
- n=100 → C = 18*100 + 100 = 1800 + 100 = 1900

First Differences:

- 550 - 100 = 450
- 1000 - 550 = 450
- 1450 - 1000 = 450
- 1900 - 1450 = 450

Perfect. So if n goes up by 25, first difference is 25 * 18 = 450.

Alternatively, if we wanted to start at 25 like Menu 1, we could do n=25,50,75,100,125 — but then we’d have to adjust. Since the table doesn’t specify starting point, and Menu 2 started at 0, I think starting at 0 is fine. But let me check the original image description — ah, in the user’s text, for Menu 3 table, it shows empty cells under “n No. of people”, no numbers filled in. So we can choose.

Actually, to match the format of Menu 1 (which starts at 25) and Menu 2 (starts at 0), perhaps for consistency in having 5 entries, let’s do for Menu 3: n = 0, 25, 50, 75, 100 — that’s 5 values, increasing by 25.

But wait — in the original problem statement, for Menu 1, n starts at 25 and goes up by 25; Menu 2 starts at 0 and goes up by 50. So for Menu 3, since it’s not specified, we can choose. However, to make the graphing easier later, maybe using the same increment as Menu 1 (25) is good.

I’ll proceed with n: 0, 25, 50, 75, 100 for Menu 3.

But let me double-check: if we start at 0, then the first entry is 100, which is the base fee — that makes sense.

So finalizing:

Menu 3 Table:

n | C | First Difference
--|---|----------------
0 | 100 |
25 | 550 | 450
50 | 1000 | 450
75 | 1450 | 450
100 | 1900 | 450

And "*n goes up by 25"

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Now, part 2a: Graph the 3 relations on the same axes.

We need to plot points for each menu.

For Menu 1: Points are (25,350), (50,600), (75,850), (100,1100), (125,1350)

For Menu 2: (0,100), (50,800), (100,1500), (150,2200), (200,2900)

For Menu 3: (0,100), (25,550), (50,1000), (75,1450), (100,1900)

Note: All three lines have the same y-intercept? Wait — Menu 1 at n=0 would be C=100, but we didn't calculate it because n started at 25. Similarly, Menu 3 at n=0 is 100. Menu 2 at n=0 is 100. So all three lines pass through (0,100)? Let's check:

Menu 1: C=10n+100 → at n=0, C=100 → yes.

Menu 2: C=14n+100 → at n=0, C=100 → yes.

Menu 3: C=18n+100 → at n=0, C=100 → yes.

So all three lines start at (0,100). That’s the base fee.

Then they rise at different slopes: 10, 14, 18.

So when graphing, we can plot from n=0 to n=200 or so.

Scale: Let’s say x-axis (n) from 0 to 200, each grid line = 25 people.

Y-axis (C) from 0 to 3000, each grid line = 250 or 500? Let’s see max cost is 2900 for Menu 2 at n=200. So maybe y-axis from 0 to 3000, each major grid = 500.

Title: "Cost vs Number of People for Three Menus"

Labels: x-axis "Number of People (n)", y-axis "Total Cost ($)"

Plot the points and draw straight lines through them.

Part 2b: Explain whether to use dashed or solid lines.

Since the number of people must be whole numbers (you can’t have half a person), technically the domain is discrete — only integer values of n. However, in many real-world contexts, especially for large gatherings, we treat it as continuous for simplicity, and the cost function is linear, so we often draw solid lines to show the relationship.

But strictly speaking, since n must be a whole number (you can’t serve 25.5 people), the graph should consist of individual points, or if connecting, perhaps dashed lines to indicate it’s not truly continuous.

However, in most school math problems like this, unless specified, they expect solid lines because the equation is linear and defined for all real n, even though practically n is integer.

But let’s think: the problem says “large gatherings”, and n is number of people — so it should be integers. Therefore, the graph should ideally be a series of points, or if connected, dashed lines to show it’s not continuous.

But looking at common practice in such textbooks, they usually draw solid lines for linear cost functions even when n is discrete, because the focus is on the linear relationship.

To be precise: since n represents count of people, it’s discrete, so we should use dashed lines or just plot points. But if we connect them, dashed lines indicate that intermediate values aren’t meaningful.

I think the expected answer is: solid lines, because the cost function is linear and we’re modeling it as a continuous relationship for simplicity.

Wait — let me reconsider. In many curricula, for situations where the variable is discrete (like number of people), they teach to use dashed lines or only plot points. For example, if you can’t have 1.5 people, then between n=1 and n=2, there’s no value, so the line shouldn’t be solid.

Upon second thought, I recall that in some standards, for discrete data, you don’t connect the dots, or you use dashed lines. But in this case, since the cost is defined for any n (even fractional, theoretically), but practically only integers, it’s ambiguous.

However, the problem says “graph the 3 relations” — and the relations are given as equations C=... which are defined for all real n. So probably, they expect solid lines.

But to be safe, let’s see what makes sense. If we plot only the points we calculated, and connect them with solid lines, it implies that for any n (even non-integer), the cost is defined, which might not be realistic, but mathematically it’s fine.

I think for this level, they want solid lines.

Actually, let me check online or recall: in typical algebra classes, for linear functions representing real-world scenarios with discrete inputs, they still often draw solid lines, noting that only integer values make sense, but the line helps visualize the trend.

Given that, I’ll go with solid lines.

But I’m not 100% sure. Another way: the problem asks to “explain whether to use dashed or solid lines”. So we need to reason.

Reasoning: Since the number of people must be a whole number (you can't have a fraction of a person attending), the actual data points are only at integer values of n. Therefore, the graph should not imply that costs exist for non-integer n. Hence, dashed lines should be used to connect the points, indicating that the relationship is only valid at discrete intervals. Alternatively, some might argue to just plot points without connecting, but if connecting, dashed is better.

I found a reference in my mind: in some textbooks, for discrete domains, they use open circles or dashed lines. For example, if n is number of items, and you can't buy half, then dashed lines.

So I think the correct answer is dashed lines.

Let me confirm with logic: if you draw a solid line from (0,100) to (25,350) for Menu 1, it suggests that at n=12.5, cost is 225, but you can't have 12.5 people, so that point isn't meaningful. Therefore, to avoid misleading, use dashed lines.

Yes, I think dashed lines is more accurate.

So for 2b: Use dashed lines because the number of people must be a whole number, so the cost is only defined at integer values of n. Connecting with dashed lines shows that the intermediate values are not practically meaningful.

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Now, compiling the final answers.

First, complete the tables.

Menu 1 Table:

n | C | First Difference
--|---|----------------
25 | 350 |
50 | 600 | 250
75 | 850 | 250
100 | 1100 | 250
125 | 1350 | 250

*n goes up by 25*

Menu 2 Table:

n | C | First Difference
--|---|----------------
0 | 100 |
50 | 800 | 700
100 | 1500 | 700
150 | 2200 | 700
200 | 2900 | 700

*n goes up by 50*

Menu 3 Table:

I chose n going up by 25, starting from 0:

n | C | First Difference
--|---|----------------
0 | 100 |
25 | 550 | 450
50 | 1000 | 450
75 | 1450 | 450
100 | 1900 | 450

*n goes up by 25*

But is starting from 0 okay? Menu 1 started from 25, but that's fine. To match the number of rows, and since Menu 2 started from 0, it's acceptable.

If the problem expects Menu 3 to start from 25 like Menu 1, then:

n=25,50,75,100,125

C=18*25+100=450+100=550

18*50+100=900+100=1000

18*75+100=1350+100=1450

18*100+100=1800+100=1900

18*125+100=2250+100=2350

First differences: 1000-550=450, etc., same.

And *n goes up by 25*

This might be better because Menu 1 also started at 25, and has 5 entries: 25,50,75,100,125.

Menu 2 started at 0 with 5 entries: 0,50,100,150,200.

So for Menu 3, to have symmetry, perhaps start at 25 and go to 125, same as Menu 1.

In the original image description, for Menu 3 table, it shows empty cells, no indication of starting point. But to be consistent with Menu 1 (which has n=25,50,75,100,125), I'll do the same for Menu 3.

So revised Menu 3:

n | C | First Difference
--|---|----------------
25 | 550 |
50 | 1000 | 450
75 | 1450 | 450
100 | 1900 | 450
125 | 2350 | 450

*n goes up by 25*

Yes, this matches Menu 1's n-values.

And first difference is 450, as 25*18=450.

Perfect.

So final tables:

Menu 1:
n: 25,50,75,100,125
C: 350,600,850,1100,1350
First Diff: 250,250,250,250

Menu 2:
n: 0,50,100,150,200
C: 100,800,1500,2200,2900
First Diff: 700,700,700,700

Menu 3:
n: 25,50,75,100,125
C: 550,1000,1450,1900,2350
First Diff: 450,450,450,450
*n goes up by 25*

For graphing, plot these points.

For 2b, use dashed lines because n must be integer.

Now, writing the final answer.

Final Answer:

Completed Tables:

Menu 1: C = 10n + 100
| n (No. of people) | C (Cost $) | First Difference |
|-------------------|------------|------------------|
| 25 | 350 | |
| 50 | 600 | 250 |
| 75 | 850 | 250 |
| 100 | 1100 | 250 |
| 125 | 1350 | 250 |
*n goes up by 25*

Menu 2: C = 14n + 100
| n (No. of people) | C (Cost $) | First Difference |
|-------------------|------------|------------------|
| 0 | 100 | |
| 50 | 800 | 700 |
| 100 | 1500 | 700 |
| 150 | 2200 | 700 |
| 200 | 2900 | 700 |
*n goes up by 50*

Menu 3: C = 18n + 100
| n (No. of people) | C (Cost $) | First Difference |
|-------------------|------------|------------------|
| 25 | 550 | |
| 50 | 1000 | 450 |
| 75 | 1450 | 450 |
| 100 | 1900 | 450 |
| 125 | 2350 | 450 |
*n goes up by 25*

2a) Graph the points from each table on the same coordinate plane. Use x-axis for number of people (n) from 0 to 200, y-axis for cost (C) from 0 to 3000. Plot the points and connect them with lines. Title: "Cost vs Number of People for Three Menus". Label axes appropriately.

2b) Use dashed lines to draw the graphs because the number of people must be a whole number (you can't have a fraction of a person), so the cost is only defined at integer values of n. Dashed lines indicate that the relationship is not continuous between those points.