Problem 13 - 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. $$(0,2),(-2,1),(3,3),(1,3),(4,4)$$

Step-by-Step Solution

Answer

The equation of the line is \(y = 0.5x + 2\)

Step 1: Create a Scatter Plot

With the help of graph paper or any graphing tool, plot each of the points: (0,2),(-2,1),(3,3),(1,3),(4,4) on a coordinate plane.

Step 2: Draw a Line of Fit

To draw a line of fit, choose two points that would be closest to all other points if the line passes through them. In this case, the points (0,2) and (4,4) seem to be the best choice. Draw a line that passes through these two points.

Step 3: Find the Equation of the Line

Use the two points to find the slope of the line. A line's slope is calculated by the formula \((y_{2}-y_{1})/(x_{2}-x_{1})\). Substituting \(x_{1} = 0, y_{1} = 2, x_{2} = 4, y_{2} =4\) into the formula, the slope (m) equals 0.5. The equation of any line on a plane can be represented as \(y = mx + c\), where m is the slope and c is the y intercept. Hence, in this case c (the y-intercept where the line cuts the y-axis) is 2, hence equation of our line is \(y = 0.5x + 2\).

Key Concepts

Line of Best Fit

When working with scatter plots, the line of best fit (sometimes called the "trend line") helps identify the relationship between two variables. Imagine you have a scatter plot with a bunch of points plotted on it. Each of these points represents an observation from your dataset. Now, the job of the line of best fit is to capture the general pattern that these points follow. A quick and easy way to draw this line is by eye-balling it. You want to place it in a position where it is as close as possible to all points on the scatter plot. It doesn’t have to pass through every point. Instead, it should reflect the trend these points suggest—sometimes above the line, sometimes below. In statistical analysis, more complex methods can be used to calculate this line, like linear regression, but for a simple exercise, repositioning by sight works well. Always pick two points that seem to form a trend and draw a line through them. This will be your line of best fit.

Equation of a Line

Once you have your line of best fit, the next step is finding the equation of that line. The equation of a line in a two-dimensional space can be expressed in the form of: \[ y = mx + c \] where:

• \( m \) represents the slope of the line.
• \( c \) is the y-intercept, which is where the line crosses the y-axis.

This equation essentially represents a straight line, defined by its slope and where it starts on the y-axis. To find the equation, you first need the slope, and then the y-intercept. In our exercise, let's pick the points (0, 2) and (4, 4) for a simple calculation. Once the slope is determined (which we'll discuss how to do next), finding \( c \), the y-intercept is simply asking where does this line meet the y-axis. In our straightforward example, it meets at 2, so our final equation was \( y = 0.5x + 2 \).

Slope Calculation

Calculating the slope is a critical step in understanding how steep the line of best fit is. Think of slope as the rate of change, or how much y changes when x increases. The method for finding the slope between two points \[ (x_1, y_1) \text{ and } (x_2, y_2) \] is by using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] The trickiest part is plugging in the correct coordinates into the formula. Let’s break this down using our chosen points from the example: (0, 2) and (4, 4). Assign generally the first coordinates as \( x_1 \) and \( y_1 \), and the second as \( x_2 \) and \( y_2 \). When you substitute these values into the formula, it looks like this:

• Substitute \( y_2 = 4 \) and \( y_1 = 2 \)
• And \( x_2 = 4 \) and \( x_1 = 0 \)

Plug these into the formula: \[ m = \frac{4 - 2}{4 - 0} = \frac{2}{4} = 0.5 \]This gives you a slope of 0.5, indicating that for each step across the x-axis, the line rises by half a step on the y-axis. Understanding slope gives you insight into how variables interact in your dataset.