91Ó°ÊÓ

Scoop has a rolling ice cream cart. He recorded his daily sales for the last seven days and the mean daytime temperature for each day. $$ \begin{aligned} &\text { Ice Cream Sales }\\\ &\begin{array}{|l|r|r|r|r|r|r|r|} \hline \text { Day } & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text { Temperature ('F) } & 83 & 79 & 75 & 70 & 71 & 67 & 62 \\ \hline \text { Sales (cones) } & 66 & 47 & 51 & 23 & 33 & 30 & 21 \\ \hline \end{array} \end{aligned} $$ a. Find the equation of the line that passes through the points \((79,47)\) and \((67,30)\). (Use the second point as the point in the point-slope form.) (a) b. Graph the data and your line from 8 a on your calculator. Sketch the result. You should have noticed in \(8 \mathrm{~b}\) that the line does not fit the data well. In fact, no two points from this data set make a good model. In \(8 \mathrm{c}-\) e you'll adjust the values of \(y_{1}\) c. Copy the table shown, and begin by changing the value of \(y_{1}\). Write two new equations, one with a larger value for \(y_{1}\) and one Value with a smaller value for \(y_{1}\). Graph each equation, and describe how the graphs compare to your original equation. (a) d. Now write two new equations that have the same values of \(x_{1}\) and \(y_{1}\) as the original, but larger and smaller values of \(b\). Graph each equation, and describe how the graphs compare to your original equation. e. Continue to adjust your values for \(y_{1}\) and \(b\) until you find a line that fits the data well. Record your final equation, Graph your equation with the data and sketch the result.

Step by step solution

01

Calculate the Slope

To find the equation of the line passing through two points, we first need to calculate the slope \( m \). The formula for the slope given two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substituting the points \((79, 47)\) and \((67, 30)\) into the formula:\[m = \frac{30 - 47}{67 - 79} = \frac{-17}{-12} = \frac{17}{12}\]

02

Use Point-Slope Form

With the slope \( m = \frac{17}{12} \) calculated, we use the point-slope form of a line's equation:\[y - y_1 = m(x - x_1)\]Substituting \((x_1, y_1) = (67, 30)\) and \( m = \frac{17}{12} \):\[y - 30 = \frac{17}{12}(x - 67)\]This is the equation of the line in point-slope form.

03

Convert to Slope-Intercept Form

Simplify the equation from step 2 to get it into slope-intercept form \( y = mx + b \):\[y - 30 = \frac{17}{12}x - \frac{17}{12} \times 67\]Calculating \(- \frac{17}{12} \times 67\):\[- \frac{17 \times 67}{12} = - \frac{1139}{12}\]Hence the equation becomes:\[y = \frac{17}{12}x - \frac{1139}{12} + 30\]Converting 30 to a fraction with a common denominator:\[- \frac{1139}{12} + \frac{360}{12} = - \frac{779}{12}\]Thus:\[y = \frac{17}{12}x - \frac{779}{12}\]

04

Graph and Analyze

Graph this line equation on your calculator along with the given data points:- Data points: \((83, 66), (79, 47), (75, 51), (70, 23), (71, 33), (67, 30), (62, 21)\).- Line: \( y = \frac{17}{12}x - \frac{779}{12} \).After plotting, you'll observe that the line does not fit the data points well as it only closely aligns with two points, and other points significantly deviate.

05

Adjusting the y-value

Duplicate the table and adjust \( y_1 = 30 \) slightly. Try \( y_1 = 35 \) and \( y_1 = 25 \) for new line equations:- For \( y_1 = 35 \), the equation is \( y - 35 = \frac{17}{12}(x - 67) \).- For \( y_1 = 25 \), the equation is \( y - 25 = \frac{17}{12}(x - 67) \).Graph these lines to see how adjusting \( y \) affects fit.

06

Adjusting the Intercept

Keep the original slope and \( x_1, y_1 = (67, 30) \), then adjust the intercept (find a new \( b \) value). Try with intercepts \( b + 5, b - 5 \) from the original line:- New equations with these intercepts should be graphed to see how adjusting \( b \) affects fit.

07

Find Best Fit

Continue adjusting \( y_1 \) and \( b \) values, replotting these adjusted lines until you find a line that reasonably fits the data pattern. This step requires iterative graphical analysis and adjustments.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point-Slope Form

The point-slope form is a way to express the equation of a line using a point on the line and the slope. It is particularly useful when you know a specific point the line passes through and the slope of the line. The general form of the point-slope equation is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) represents the known point and \( m \) is the slope.

To apply this in the exercise, we used the points \((79, 47)\) and \((67, 30)\). First, the slope \( m \) was calculated using the formula for slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This gave a slope of \( \frac{17}{12} \).

We then substituted the point \((67, 30)\) into the point-slope form, yielding the equation \( y - 30 = \frac{17}{12}(x - 67) \). This method simplifies writing equations and is a critical tool in algebra to create linear models directly from data points.

Slope-Intercept Form

The slope-intercept form of a line is one of the most straightforward ways to express a linear equation. This form is written as \( y = mx + b \), where \(m\) is the slope of the line, and \(b\) is the y-intercept—the point where the line crosses the y-axis.

To convert from point-slope to slope-intercept form, algebraic manipulation is used to simplify the equation into the \(y = mx + b\) structure. In the exercise, after finding the point-slope form (\( y - 30 = \frac{17}{12}(x - 67) \)), the equation was simplified by distributing the \( \frac{17}{12} \) and solving for \( y \). This involved handling fractions and combining terms leading to the slope-intercept form: \( y = \frac{17}{12}x - \frac{779}{12} \).

This form is helpful because it immediately provides the slope and y-intercept, making it easier to graph and compare with other lines.

Graphing Data

Graphing is a fundamental skill in understanding the relationship between data sets. By plotting data points on a graph, you can visually interpret the trends and relationships. In this exercise, the graph depicts the relationship between temperature and ice cream sales.

When graphing data with a line equation, such as \( y = \frac{17}{12}x - \frac{779}{12} \), you draw a line that represents the equation in the context of the data points provided and observe how well it fits the pattern of data points. Ideally, the line should reflect the overall trend in the data, minimizing the distance between the data points and the line itself.

This step not only involves initial plotting but also requires adjusting parameters such as the slope or intercept for the best fit, illustrating how small changes can influence the accuracy of the model with real-world data. These adjustments, like those made for \( y_1 \) and \( b \) in this exercise, help in finding a model that aligns closely with observed data, providing deeper insight into the linear relationship between variables.