Which Line Fits Best?

Mia counts her push-ups in one minute after each week of training. Week 1 to 8: $6, 6, 9, 12, 12, 16, 16, 19$. Three students draw a line by eye: - Line A: $y = 2x + 3$ - Line B: $y = 2x + 7$ - Line C: $y = 0.5x + 9$ Which line is the best line of best fit?

Check line B: Line B gives $9, 11, 13, 15, 17, 19, 21, 23$ for weeks 1 to 8. Every data point is lower. = All 8 points are below line B: it is too high

Check line C: Line C gives $9.5, 10, 10.5, 11, 11.5, 12, 12.5, 13$. Weeks 1 to 3 are below it, weeks 4 to 8 above it. = Points below at the left end, above at the right end: line C is too flat

Check line A: Line A gives $5, 7, 9, 11, 13, 15, 17, 19$. Above: weeks 1, 4, 6. Below: weeks 2, 5, 7. On the line: weeks 3, 8. = 3 above, 3 below, spread along the whole line

Decide: Only line A has the points balanced on both sides along its whole length and follows the upward trend. = Line A is the best fit

Answer: Line A, $y = 2x + 3$, is the best line of best fit. Line B is too high and line C is too flat.

Equation From Two Points on the Line

Mia's line of best fit passes through the grid points $(0, 3)$ and $(5, 13)$. Find its equation.

Find the change in y and in x: $13 - 3 = 10$ and $5 - 0 = 5$ = Rise $10$, run $5$

Calculate the slope: $m = \frac{10}{5} = 2$ = $m = 2$

Find the intercept: The point $(0, 3)$ is on the y-axis, so $b = 3$ = $b = 3$

Write and check the equation: $y = 2x + 3$. Check with $(5, 13)$: $2 \times 5 + 3 = 13$ = $y = 2x + 3$

Answer: The line of best fit is $y = 2x + 3$.

A Falling Trend

A café plots the outside temperature $x$ (°C) against the number of hot drinks sold $y$. The line of best fit drawn by eye passes through $(10, 80)$ and $(25, 35)$. Find its equation and explain the slope.

Calculate the slope: $m = \frac{35 - 80}{25 - 10} = \frac{-45}{15} = -3$ = $m = -3$

Use one point to find b: $b = y_1 - m x_1 = 80 - (-3)(10) = 80 + 30 = 110$ = $b = 110$

Write the equation: $y = -3x + 110$ = $y = -3x + 110$

Check with the other point: $-3 \times 25 + 110 = -75 + 110 = 35$ = The line passes through $(25, 35)$

Interpret the slope: The slope is negative: for each degree warmer, the line drops by 3 drinks = About 3 fewer hot drinks per extra °C

Answer: $y = -3x + 110$: for every degree warmer, the café sells about 3 fewer hot drinks.

What Do Slope and Intercept Mean?

For used bikes, the line of best fit between age $x$ (years) and price $y$ (dollars) is $y = -40x + 300$. The data covers bikes from 1 to 6 years old. Explain the slope and the intercept.

Read the slope: $m = -40$, measured in dollars per year = Slope $-40$

Interpret the slope: Each extra year of age, the predicted price goes down by 40 dollars = A bike loses about 40 dollars per year

Read the intercept: $b = 300$, the value of $y$ when $x = 0$ = Intercept $300$

Interpret the intercept: Age $0$ means a new bike. It is just outside the data (1 to 6 years), so it is a rough estimate = A new bike costs about 300 dollars

Answer: The slope says a used bike loses about 40 dollars per year. The intercept says a new bike costs about 300 dollars.

Using the Mean Point

A kiosk records hours of sunshine $x$ and ice creams sold $y$ on five days: $(1, 4)$, $(3, 7)$, $(4, 10)$, $(6, 12)$, $(6, 17)$. A student drew the line $y = 2x + 1$. Check it against the mean point and fix it, keeping the slope $2$.

Find the mean of x: $\bar{x} = \frac{1 + 3 + 4 + 6 + 6}{5} = \frac{20}{5} = 4$ = $\bar{x} = 4$

Find the mean of y: $\bar{y} = \frac{4 + 7 + 10 + 12 + 17}{5} = \frac{50}{5} = 10$ = $\bar{y} = 10$

Test the student's line at x = 4: $2 \times 4 + 1 = 9$, but $\bar{y} = 10$ = The line misses $(4, 10)$ by 1

Move the line through the mean point: $b = \bar{y} - m\bar{x} = 10 - 2 \times 4 = 2$ = $y = 2x + 2$

Answer: The mean point is $(4, 10)$. The improved line is $y = 2x + 2$, which passes through it.

Mistake: Drawing the line from the first data point to the last one

Why: The first and last points can be unusual. A line through them often has most points on one side.

Correct: Ignore single points and look at the whole cloud: about half the points above, half below. The line may pass through no point at all.

Mistake: Balancing the number of points but not following the trend

Why: A line that is too flat can have 4 points above and 4 below, but all the low ones sit at the left end and the high ones at the right end.

Correct: Check both ends: near each end of the line there should be points above and below it.

Mistake: Finding the slope from two data points instead of two points on the line

Why: Data points are usually not on the line, so their slope is not the slope of your line.

Correct: Read two points that lie on your drawn line, far apart and on grid crossings, then use $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Mistake: Reading the slope as a value of y instead of a change

Why: In $y = 2x + 3$ for weeks and push-ups, $2$ is not "2 push-ups". It is the change per week.

Correct: Say the slope with both units: about 2 more push-ups per week of training.

Training for a Race

Coaches draw a trend line through an athlete's times to see how much faster they get each week.

A runner's 5 km times drop from about 30 minutes to about 22 minutes over a few weeks. The trend line shows the steady improvement even on bad days.

Lemonade Stand

Small sellers plot temperature against sales to plan how much to prepare.

On hot days a stand sells many more cups. A trend line through a summer of data shows about how many extra cups each degree brings.

Plant Growth Experiment

In science class, students measure a plant every few days and use a trend line to describe its growth rate.

Measurements are never perfect, so the heights scatter a little. The line of best fit gives one growth rate in centimeters per day.

Q: Is there only one correct line of best fit?

A: By eye, no. Two students can draw slightly different lines and both be good. Their slopes and intercepts will be close. Only the least-squares line, from the Linear Regression lesson, is one exact line.

Q: Does the line have to go through the origin?

A: No. The line goes wherever the data points it. Only force it through $(0, 0)$ if the situation really starts at zero and the data agrees.

Q: Should I count points that lie exactly on the line?

A: Points on the line count for neither side. Balance the points above against the points below.

Q: When should I not draw a line of best fit?

A: When the scatter plot shows no trend, or when the points follow a clear curve. A straight line only makes sense for a linear pattern.