Comprehensive worksheet guiding students through finding the line of best fit using graphing calculators.

Here is the step-by-step solution to the problems on the worksheet.

Part 1: Fill in the Blanks & Multiple Choice

1. Definition of Line of Best Fit
* Question: line of best fit - a line that best \_\_\_\_\_\_\_\_\_ the data on a scatter plot.
* Answer: represents (or sometimes "fits").
* *Reasoning:* The line represents the general trend of the data points.

2. Multiple Choice Example
* Question: Which function would be the best fit for the scatter plot?
* Looking at the graph, the dots go up from left to right. This means the slope must be positive. We can eliminate options (a) and (d) because they have negative slopes ($-x$ and $-2x$).
* Now we look at where the line crosses the y-axis (the vertical axis). The dots are below the x-axis when $x=0$. This means the y-intercept must be negative. Both (b) and (c) have negative intercepts ($-2$ and $-1$).
* Let's check the slope more closely. The points rise gradually. A slope of $\frac{3}{4}$ (0.75) is steeper than $\frac{3}{5}$ (0.6). Looking at the grid, for every 5 units right, it goes up about 3 units. Also, at $x=0$, the imaginary line seems to hit around $y=-1$ or slightly lower.
* Let's test option (c) $y = \frac{3}{5}x - 1$. If $x=5$, $y = 3 - 1 = 2$. On the graph, at $x=5$, the dot is near $y=2$.
* Let's test option (b) $y = \frac{3}{4}x - 2$. If $x=4$, $y = 3 - 2 = 1$. On the graph, at $x=4$, the dot is near $y=1$.
* Actually, looking closer at the origin $(0,0)$, the cluster starts around $x=-2, y=-2$ and goes to $x=5, y=2$.
* Rise = $2 - (-2) = 4$. Run = $5 - (-2) = 7$. Slope $\approx \frac{4}{7} \approx 0.57$.
* $\frac{3}{5} = 0.6$. $\frac{3}{4} = 0.75$. $0.6$ is closer to $0.57$.
* Let's check the y-intercept again. If slope is $0.6$ and it passes through $(5,2)$: $2 = 0.6(5) + b \rightarrow 2 = 3 + b \rightarrow b = -1$.
* Therefore, c. $y = \frac{3}{5}x - 1$ is the best fit.

3. Definition of Linear Regression
* Question: linear regression - a method for finding the \_\_\_\_\_ of best \_\_\_\_\_.
* Answer: line, fit.

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Part 2: Graphing Calculator Problem

Data Table:
* L1 (Hours): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
* L2 (GPA): 2.2, 2.4, 2.5, 2.8, 2.9, 3.0, 3.2, 3.3, 3.5, 3.7, 4.0

Step 1: Find the Line of Best Fit Equation
Using the linear regression steps provided (LinReg(ax+b)):
* The calculator calculates the slope ($a$) and y-intercept ($b$).
* Sum of $x = 55$, Sum of $y = 33.5$, Count ($n$) = 11.
* Mean of $x = 5$, Mean of $y \approx 3.045$.
* After performing the regression calculation:
* Slope ($a$) $\approx 0.177$
* Y-intercept ($b$) $\approx 2.155$
* So, the equation is approximately: $y = 0.177x + 2.155$

*(Note: Depending on rounding, you might see $y = 0.18x + 2.15$)*

Step 2: Estimate GPA for 2.5 Hours
The question asks to estimate the GPA for a student who studies 2.5 hours per week. We plug $x = 2.5$ into our equation.

$$y = 0.177(2.5) + 2.155$$

1. Multiply slope by hours:
$$0.177 \times 2.5 = 0.4425$$
2. Add the y-intercept:
$$0.4425 + 2.155 = 2.5975$$

Rounding to one decimal place (since the GPAs in the table are given to one decimal place):
$$2.5975 \approx 2.6$$

Let's double-check with the rounded equation $y = 0.18x + 2.15$:
$$y = 0.18(2.5) + 2.15$$
$$y = 0.45 + 2.15$$
$$y = 2.60$$

Both methods give us 2.6.

Final Answer:
Fill in the blanks:
1. represents
2. c. $y = \frac{3}{5}x - 1$
3. line, fit

Graphing Calculator Problem:
Equation: $y = 0.177x + 2.155$ (or $y = 0.18x + 2.15$)
Estimated GPA for 2.5 hours: 2.6