Problem 55 - A Graphical Approach to College Algebra

Question

Match each equation with the graph that it most closely resembles. $$y=-3 x-6$$

Step-by-Step Solution

Answer

The graph is a straight line with a slope of -3, crossing the y-axis at -6.

Step 1: Determine the Type of Equation

The given equation \( y = -3x - 6 \) is a linear equation because it is in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Identify the Slope

In the equation \( y = -3x - 6 \), the slope \( m \) is equal to \(-3\). This indicates that for every unit increase in \( x \), \( y \) decreases by 3 units. This results in a line that slopes downwards from left to right.

Step 3: Determine the Y-Intercept

The y-intercept \( b \) in the equation \( y = -3x - 6 \) is \(-6\). This means the line crosses the y-axis at the point (0, -6).

Step 4: Sketch the Graph

Start by plotting the y-intercept at (0, -6) on the coordinate plane. From this point, use the slope \(-3\) (rise over run = -3/1) to find another point. Move down 3 units and right 1 unit from (0, -6) to plot the next point at (1, -9).

Step 5: Draw the Line

Draw a straight line through the points (0, -6) and (1, -9). This line should extend in both directions, maintaining a consistent slope of \(-3\).

Key Concepts

Slope

The slope of a line in a linear equation is a key factor that determines the line's angle and direction. In the equation of the form \( y = mx + b \), the letter \( m \) represents the slope. Simply put, the slope shows how steep the line is. A slope of \(-3\) means that for every 1 unit you move to the right along the x-axis, you move down 3 units on the y-axis. Thus, you will notice a line that slopes downwards at a sharp angle from left to right when plotted on a graph.

• A positive slope means the line moves up as you go from left to right.
• A negative slope, like \(-3\), means the line goes down.
• Larger absolute values of the slope indicate steeper lines.

Y-intercept

The y-intercept is where the line crosses the y-axis on a graph. You can find it easily from the linear equation \( y = mx + b \), where \( b \) represents the y-intercept. For the equation \( y = -3x - 6 \), the y-intercept is \(-6\). This means the line will intersect the y-axis at the point (0, -6).
Understanding the y-intercept makes graphing linear equations much simpler because it gives you a starting point for your line.

• The y-intercept is always located at \( x = 0 \).
• It helps in quickly drawing the graph by providing a starting location.
• The value tells you how high or low the line starts on the y-axis.

Graphing

Graphing a linear equation involves transforming the equation into a visual format, making it easier to see the relationship between \( x \) and \( y \). Start by identifying the y-intercept from the equation; for \( y = -3x - 6 \), this is \(-6\). You plot this point where the line crosses the y-axis on the graph at (0, -6).
Next, use the slope to determine the direction of the line. With a slope of \(-3\), move 1 unit to the right and 3 units down to find another point. Plot this point at (1, -9).
Connect these two points with a straight line, which represents all possible points (x, y) that satisfy the equation. This line will continue infinitely in both directions, maintaining the same slope.

• Always begin graphing with the y-intercept to simplify the process.
• Use the slope consistently to plot additional points.
• Ensure the line is straight and extends in both directions to properly represent the equation.