Problem 15 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) create a scatter plot of the data, (b) draw a line of fit that passes through two of the points, and (c) use the two points to find an equation of the line. $$(0,7),(3,2),(6,0),(4,3),(2,5)$$

Step-by-Step Solution

Answer

The resulting line equation from the two points (0,7) and (6,0) will be \(y = -\frac{7}{6}x + 7\).

Step 1: Creating the Scatter Plot

Plot the given points on a graph. This will give a visual representation of the data. The points are (0,7), (3,2), (6,0), (4,3), and (2,5).

Step 2: Draw the Line of Best Fit

Drawing a best-fit line requires you to visualize a line that best represents the distribution of all data points. The line does not necessarily have to pass through every single point but should fall in the middle of the points. Let's say for this exercise, the best fit line passes through points (0,7) and (6,0).

Step 3: Find the Equation of the Line

To find the equation of the best fit line, use the formula for the equation of a line in the slope-intercept form, which is \(y = mx + c\). 'm' is the slope of the line and can be found using the formula \((y2 - y1) / (x2 - x1)\), where \((x1, y1)\) and \((x2, y2)\) are any two points through which the line passes, and 'c' is the y-intercept, where the line crosses the y-axis. In this case, our chosen points are (0,7) and (6,0). By plugging in these values, we can calculate 'm' and 'c'.

Key Concepts

Line of Best Fit

The "line of best fit," also known as the "trend line," is a straight line that represents the general direction that a set of points tend to follow in a scatter plot. It helps to uncover the underlying trend of the data and can be used to make predictions. When drawing a line of best fit, the idea is to have the line as close as possible to all data points.

• The line doesn't need to go through all the points or even any points.
• It should evenly pass through the scattered points, showing the trend clearly.

For example, in the exercise, the best fit line passes through the points (0,7) and (6,0). Although it doesn’t touch all points, it shows the general alignment of the data.

Equation of a Line

An equation of a line in mathematical terms can describe the particular line on a graph. This is crucial because it provides a formula that represents the relationship between the x (horizontal) and y (vertical) coordinates of every point on that line. To find the equation, we need at least two points through which the line passes.

• These points provide the necessary information to calculate the slope and y-intercept.
• By using these components, we can formulate the line's equation.

The method generally involves finding two key components of the line, the slope and the y-intercept, which form the base of the line's equation.

Slope-Intercept Form

The slope-intercept form is a mathematical way to express the equation of a straight line. It is given by the formula:\[ y = mx + c \]Where:

• m stands for the slope of the line. It indicates the line's steepness or inclination.
• c is the y-intercept, which is the point where the line crosses the y-axis.

Calculating the slope is done by using any two chosen points on the line. For the slope calculus, apply:\[ m = \frac{(y2 - y1)}{(x2 - x1)} \]The y-intercept 'c' is easily found using one of the points already known and substituting into the slope-intercept form. This form allows you to quickly write the equation once 'm' and 'c' are known, making it an essential tool for understanding linear relationships between variables.

Data Visualization

Data visualization refers to the graphical representation of information and data. It enables us to see patterns, trends, and correlations that might not be immediately obvious from data in their raw form.

• Scatter plots are a common form of data visualization. They allow us to display values for typically two variables for a set of data.
• By plotting individual data points, we can start to see the "shape" of the data distribution.

In the exercise example, plotting points such as (0,7), (3,2), (6,0), (4,3), and (2,5) on a graph will give a visual representation, helping to quickly analyze the data. This visual representation is key to creating a line of best fit and understanding the relationship dynamics between variables at a glance.