Problem 102 - College Algebra Essentials

Question

Use a graphing utility to graph each equation.Then use the TRACE feature to trace along the line and find the coordinates of two points Use these points to compute the line's slope. Check your result by using the coefficient of \(x\) in the line's equation. $$y=-\frac{1}{2} x-5$$

Step-by-Step Solution

Answer

The slope of the line given by the equation \(y=-\frac{1}{2}x-5\) is \(-\frac{1}{2}\)

Step 1: Plotting the Equation

Plot the line of the equation \(y=-\frac{1}{2}x-5\) on the graphing utility. Look at the plot and identify two points on the line. Typically, the points where the line intersects the x and y axes are chosen. These are obvious points and they make the calculation easier. These points can be found via tracing along the plotted line.

Step 2: Calculating the Slope

Once the coordinates of two points are determined, use the slope formula, which is \[m = \frac{y_2 - y_1}{x_2 - x_1}\] where \(m\) is the slope, \(y_2\) and \(y_1\) are the y-coordinates of the two points and \(x_2\) and \(x_1\) are the x-coordinates of the two points. So, find the vertical change (subtract the y-coordinate of the first point from the y-coordinate of the second point, \(y_2 - y_1\)), and the horizontal change (subtract the x-coordinate of the first point from the x-coordinate of the second point, \(x_2 - x_1\)). Then divide the vertical change by the horizontal change.

Step 3: Checking your Result

Compare the result of your slope calculation to the coefficient of \(x\) from the original equation, which is \(-\frac{1}{2}\). If they match, the work is done correctly.

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line is a fundamental concept in coordinate geometry. The slope measures how steep a line is. To calculate the slope, we use the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \], where:

• \(m\) represents the slope.
• \(y_2\) and \(y_1\) are the y-coordinates of the two points.
• \(x_2\) and \(x_1\) are the x-coordinates of the two points.

Start by finding two points on the line, such as where the line crosses the x-axis or y-axis. By substituting these coordinates into the slope formula, you can find how much the line moves vertically for each horizontal step. Compare the result to the coefficient of \(x\) in the equation. If they match, you've done it right! Understanding this concept is crucial for interpreting and graphing lines.

Graphing Calculator

A graphing calculator is a powerful tool that helps visualize mathematical equations. By entering an equation like \( y = -\frac{1}{2}x - 5 \), the calculator can plot the line, showing you its shape and position.To find specific points on the line, use the TRACE feature. This allows you to move along the line and see the exact coordinates of different points. It's especially useful for identifying key points like intercepts, which assist in calculating the slope effectively.Why use a graphing calculator?

• It provides a quick and accurate way to plot complex equations.
• Visualizations help reinforce understanding by showing real-world applications of math.
• Interactive features enhance the learning experience.

Coordinate Geometry

Coordinate geometry, or analytic geometry, combines algebra and geometry to describe geometric principles using a coordinate system. This makes analyzing and proving geometric theorems much easier.Key Elements:

• Coordinates: Every point on a plane is represented by an ordered pair \((x, y)\).
• Lines and Slopes: Equations like \( y = -\frac{1}{2}x - 5 \) represent lines, and the slope indicates the angle of the line relative to the x-axis.
• Intercepts: The x-intercept is where the line crosses the x-axis (\(y=0\)), and the y-intercept is where it crosses the y-axis (\(x=0\)).

By understanding these concepts, you gain insights into the graphical representation and analysis of different geometric shapes. It's a bridge between numeric and spatial understanding.

Equation of a Line

The equation of a line in two dimensions, such as \( y = -\frac{1}{2}x - 5 \), is a mathematical representation of a straight line. It shows the relationship between the x and y coordinates.Forms of Linear Equations:

• Slope-Intercept Form: \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept. This form makes it easy to plot a line and understand its characteristics.
• Point-Slope Form: Used when you know a point on the line and the slope.
• Standard Form: \( Ax + By = C \), useful for certain calculations.

The slope-intercept form is often preferred for its simplicity in graphing. It gives a clear picture of how the slope and intercept define the line's path.