Question

the scatterplot shows the relationship between two variables, x and y, for data set e. a line of best fit is shown. data set f is created by multiplying the y - coordinate of each data point from data set e by 3.9. which of the following could be an equation of a line of best fit for data set f?
a) $y = 46.8 + 5.9x$
b) $y = 46.8 + 1.5x$
c) $y = 12 + 5.9x$
d) $y = 12 + 1.5x$

Explanation:

Step1: Analyze Data Set E's Line

For data set E, the line of best fit has a negative slope? Wait, no, wait the axes: x is horizontal (from 0 to 20) and y is vertical (from 0 to 50). Wait, the scatter plot: when x increases, y decreases? Wait no, looking at the grid, x-axis is labeled with 0,4,8,12,16,20 (horizontal), y-axis 0,10,20,30,40,50 (vertical). The line goes from (x=0, y≈46.8) to (x=20, y≈12)? Wait no, maybe I misread. Wait the points: when x is 0, y is around 46.8? Wait no, the first point (x=2, y≈45), then x=4, y≈42, x=6, y≈39, x=8, y≈36, x=10, y≈33, x=12, y≈30, x=14, y≈27, x=16, y≈24, x=18, y≈21, x=20, y≈18? Wait no, maybe the line of best fit for E: let's calculate slope. Let's take two points. When x=0, y≈46.8 (intercept), and when x=20, y≈12? Wait no, 46.8 - 12 = 34.8 over 20, slope ≈ -1.74? No, the options have positive slopes. Wait, maybe I mixed up x and y axes. Wait the problem says "multiplying the y - coordinate of each data point from data set E by 3.9" to get F. Wait, no, wait the problem statement: "Data set F is created by multiplying the y - coordinate of each data point from data set E by 3.9". Wait, no, wait the original: "multiplying the y - coordinate of each data point from data set E by 3.9". Wait, no, maybe it's a typo, or maybe I misread. Wait the line of best fit for E: let's check the options. The line of best fit for E: let's see, the equation for E is y = 46.8 - 1.5x? No, the options for E's line? Wait no, the question is about F, which is E's y - coordinates multiplied by 3.9? Wait no, wait the problem says: "Data set F is created by multiplying the y - coordinate of each data point from data set E by 3.9". Wait, no, that would make y larger, but the options for F have smaller intercepts? Wait, maybe it's a mistake, or maybe it's multiplying x? No, the problem says y - coordinate. Wait, no, maybe I misread: "multiplying the y - coordinate of each data point from data set E by 3.9" – no, that would scale y by 3.9, but the options for F have intercepts 12 or 46.8, and slopes 1.5 or 5.9. Wait, let's think about the line of best fit. The line of best fit for E: let's assume the line for E is y = 46.8 - 1.5x (since when x=0, y=46.8, and slope negative). But when we multiply y by 3.9? No, that can't be. Wait, maybe it's dividing? No, the problem says multiplying. Wait, no, maybe the problem is "multiplying the x - coordinate"? No, the problem says y - coordinate. Wait, maybe there's a mistake, but let's look at the options. The line of best fit for E: let's take two points. Let's say when x=0, y=46.8 (intercept), and when x=20, y=12 (since 46.8 - 1.5*20 = 46.8 - 30 = 16.8, no. Wait 46.8 - 5.9*20 = 46.8 - 118 = negative, which is wrong. Wait, maybe the line for E is y = 46.8 - 1.5x (slope -1.5) or y = 46.8 - 5.9x (slope -5.9). But when we create F by multiplying y by 3.9? No, that would be y_F = 3.9*y_E. So if y_E = 46.8 - 1.5x, then y_F = 3.9*(46.8 - 1.5x) = 3.9*46.8 - 3.9*1.5x ≈ 182.52 - 5.85x, which is not an option. So maybe it's dividing y by 3.9? Then y_F = y_E / 3.9. So if y_E = 46.8 - 1.5x, then y_F = (46.8)/3.9 - (1.5/3.9)x = 12 - 0.38x, no. If y_E = 46.8 - 5.9x, then y_F = 46.8/3.9 - 5.9/3.9x = 12 - 1.5x, no. Wait, maybe the problem is "multiplying the x - coordinate by 3.9"? No, the problem says y - coordinate. Wait, maybe I have the axes reversed. Let's check the graph: x - axis is vertical (0 to 20) and y - axis is horizontal (0 to 50)? No, the graph shows x - axis (horizontal) labeled 0,4,8,12,16,20 and y - axis (vertical) labeled 0,10,20,30,40,50. The line is decreasing from left to right, so slope is negative. But the options for F have positive slopes. This is confusing. Wait, maybe the problem is "multiplying the y - coordinate by 0.25" (since 46.8*0.25≈12, 1.5*0.25≈0.38, no). Wait, the options for F: A) y=46.8+5.9x, B) y=46.8+1.5x, C) y=12+5.9x, D) y=12+1.5x. Wait, the intercept for F should be (intercept of E) * 3.9? No, 46.8*3.9≈182, which is not an option. So maybe it's dividing the intercept by 3.9: 46.8 / 3.9 = 12. Ah! That makes sense. So if we divide the y - coordinate of E by 3.9 to get F (maybe a typo in the problem, saying multiply instead of divide), then the intercept of E is 46.8, so intercept of F is 46.8 / 3.9 = 12. Now, the slope: the slope of E's line: let's say the slope of E is -1.5 (since 46.8 - 1.5x, and when we divide y by 3.9, the new slope would be (-1.5)/3.9? No, that's not matching. Wait, no, maybe the line of best fit for E has a positive slope? Wait, maybe I had the axes reversed. Let's assume x is vertical and y is horizontal? No, standard is x horizontal, y vertical. Wait, maybe the graph is plotted with x on the vertical and y on the horizontal. Let's try that: x - axis (vertical) 0 to 20, y - axis (horizontal) 0 to 50. Then the line is increasing from bottom to top, so slope positive. So when x=0 (vertical), y=12 (horizontal), and when x=20 (vertical), y=46.8 (horizontal). Then the slope would be (46.8 - 12)/20 = 34.8/20 = 1.74, close to 1.5 or 5.9. Wait, 5.9 is steeper. Now, if we multiply the y - coordinate (horizontal) by 3.9? No, this is too confusing. Wait, let's look at the intercepts. The intercept for F: if E's intercept is 46.8, and F's intercept is 12, then 46.8 / 3.9 = 12. So that's a division by 3.9. So the line of best fit for E is y = 46.8 + 1.5x (slope positive), and when we divide y by 3.9, we get y = (46.8 + 1.5x)/3.9 = 12 + (1.5/3.9)x ≈ 12 + 0.38x, no. If E's line is y = 46.8 + 5.9x, then dividing by 3.9 gives y = 12 + (5.9/3.9)x ≈ 12 + 1.5x. Ah! 5.9 / 3.9 ≈ 1.51, which is approximately 1.5. So that matches option D: y = 12 + 1.5x? No, wait 5.9/3.9≈1.51, so if E's line is y = 46.8 + 5.9x, then F's line (dividing y by 3.9) is y = 12 + 1.5x? Wait, no, the problem says "multiplying the y - coordinate by 3.9", but that would be y = 3.9*(46.8 + 5.9x) = 182.52 + 22.91x, which is not an option. So there must be a mistake in the problem statement, and it's actually dividing the y - coordinate by 3.9. Then, E's line: y = 46.8 + 5.9x, F's line: y = (46.8 + 5.9x)/3.9 = 12 + (5.9/3.9)x ≈ 12 + 1.5x. Wait, 5.9 divided by 3.9 is approximately 1.51, which is 1.5. So the equation for F would be y = 12 + 1.5x? No, wait 46.8/3.9=12, 5.9/3.9≈1.5, so yes. So the correct option is D? Wait no, let's check again. Wait, the problem says "multiplying the y - coordinate of each data point from data set E by 3.9" to get F. So if E's point is (x, y), F's point is (x, 3.9y). Then the line of best fit for F would be y = 3.9*(line of best fit for E). So if E's line is y = 12 + 1.5x (intercept 12, slope 1.5), then F's line would be y = 3.9*12 + 3.9*1.5x = 46.8 + 5.85x ≈ 46.8 + 5.9x, which is option A. Wait, now I'm confused. Let's start over.

1. Let the line of best fit for E be \( y_E = a + bx \).
2. Data set F is created by \( y_F = 3.9 \times y_E \) (multiplying y - coordinate by 3.9).
3. So the line of best fit for F is \( y_F = 3.9(a + bx)=3.9a + 3.9bx \).
4. Now, look at the options for F:
- Option A: \( y = 46.8 + 5.9x \)
- Option B: \( y = 46.8 + 1.5x \)
- Option C: \( y = 12 + 5.9x \)
- Option D: \( y = 12 + 1.5x \)
5. From the graph, the line of best fit for E: when x = 0, \( y_E \) (intercept) is such that \( 3.9 \times y_E \) (intercept for F) is either 46.8 or 12.
- If intercept for F is 46.8, then \( y_E \) (intercept for E) is \( 46.8 / 3.9 = 12 \).
- If intercept for F is 12, then \( y_E \) (intercept for E) is \( 12 / 3.9 \approx 3.08 \), which doesn't match the graph.
6. So intercept for E is 12, so intercept for F is \( 3.9 \times 12 = 46.8 \).
7. Now, the slope: \( 3.9 \times b \) (slope for F) should be either 5.9 or 1.5.
- If slope for F is 5.9, then slope for E is \( 5.9 / 3.9 \approx 1.51 \approx 1.5 \).
- If slope for F is 1.5, then slope for E is \( 1.5 / 3.9 \approx 0.38 \), which doesn't match the graph (the line for E is steeper).
8. So slope for E is approximately 1.5, so slope for F is \( 3.9 \times 1.5 = 5.85 \approx 5.9 \).
9. Therefore, the equation for F is \( y = 3.9 \times (12 + 1.5x)=46.8 + 5.85x \approx 46.8 + 5.9x \), which is option A.

Wait, now I see my mistake earlier. I thought F was dividing, but it's multiplying. So E's line is \( y_E = 12 + 1.5x \), then F's line is \( y_F = 3.9 \times (12 + 1.5x)=46.8 + 5.85x \approx 46.8 + 5.9x \), which is option A.

Step2: Verify with Slope and Intercept

- Intercept: E's intercept is 12, F's intercept is \( 12 \times 3.9 = 46.8 \), which matches option A and B.
- Slope: E's slope is 1.5, F's slope is \( 1.5 \times 3.9 = 5.85 \approx 5.9 \), which matches option A (slope 5.9) and not B (slope 1.5).

So the equation for F is \( y = 46.8 + 5.9x \), which is option A.

Answer:

A. \( y = 46.8 + 5.9x \)