Problem 78 - A Graphical Approach to College Algebra

Question

Sketch by hand the graph of the line passing through the given point and having the given slope. Label Through \((-2,-3), m=-\frac{3}{4}\)

Step-by-Step Solution

Answer

The line equation is \( y = -\frac{3}{4}x - \frac{9}{2} \). Plot points \((-2,-3)\) and \((2,-6)\) on the graph, then draw the line.

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line's equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For the given exercise, the slope \( m \) is \(-\frac{3}{4}\). Next, you'll need to find \( b \).

Step 2: Insert the Given Point

Insert the coordinates of the point \((-2,-3)\) into the equation \( y = mx + b \). This helps us to find the y-intercept \( b \). So, \( -3 = -\frac{3}{4}(-2) + b \).

Step 3: Solve for the y-Intercept (b)

First, calculate \(-\frac{3}{4} \times (-2) = \frac{6}{4} = \frac{3}{2}\). Now substitute back into the equation: \( -3 = \frac{3}{2} + b \). Simplifying gives \( b = -3 - \frac{3}{2}\). Convert \(-3\) to \(-\frac{6}{2}\) so you can subtract: \( b = -\frac{6}{2} - \frac{3}{2} = -\frac{9}{2} \).

Step 4: Write the Equation

With the slope and y-intercept found, write the equation of the line. With \( m = -\frac{3}{4} \) and \( b = -\frac{9}{2} \), the equation of the line is \( y = -\frac{3}{4}x - \frac{9}{2} \).

Step 5: Plot the Given Point

On a coordinate plane, locate and plot the given point \((-2,-3)\) by moving 2 units to the left of the origin and then 3 units down.

Step 6: Use the Slope to Plot a Second Point

From the point \((-2,-3)\), use the slope \(-\frac{3}{4}\). Move 4 units to the right (the denominator) and then 3 units down (the numerator) for a slope of \(-\frac{3}{4}\), arriving at \((2,-6)\).

Step 7: Draw the Line

Draw a straight line passing through both points, \((-2,-3)\) and \((2,-6)\), using a ruler. Extend the line across the coordinate plane and label it.

Key Concepts

Line Equation

A line equation represents the relationship between the x and y coordinates of a line on the coordinate plane. The most widely used form is the slope-intercept form, which is expressed as \( y = mx + b \). Here, \( m \) stands for the slope of the line, while \( b \) is the y-intercept. In our example, the line equation is \( y = -\frac{3}{4}x - \frac{9}{2} \). This equation allows us to easily graph the line by using its slope and y-intercept.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we plot points using their x and y coordinates. It consists of two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Each point on this plane is designated by an ordered pair \((x, y)\). For instance, the point \((-2, -3)\) in our problem is located by moving 2 units to the left from the origin along the x-axis and 3 units down on the y-axis. The coordinate plane is crucial for visualizing geometric relationships and solving problems in algebra.

Point Plotting

Point plotting involves marking a specific location on the coordinate plane, guided by its x and y coordinates. To plot the point \((-2, -3)\), start at the origin, which is point \((0, 0)\), located at the intersection of both axes. From there, move 2 units left (for the x-coordinate) and 3 units down (for the y-coordinate). Each plotted point aids in constructing the graph of a line and can be used to verify the accuracy of the line equation. Moreover, multiple points plotted correctly define the path of the line in context.

Y-Intercept

The y-intercept is a critical feature of a line, as it denotes where the line crosses the y-axis. In the equation \( y = mx + b \), \( b \) represents this intercept. For our line, the y-intercept is \(-\frac{9}{2}\), meaning it intersects the y-axis at this point. Understanding the y-intercept helps in quickly drawing the line on a coordinate plane, because knowing this point allows for easy calculation of the line's other points using its slope. A correct y-intercept ensures the line is drawn following the accurate path as represented by its equation.