Problem 9 - Algebra Form and Function

Question

Graph the equation. $$ y=-\frac{2}{3} x-4 $$

Step-by-Step Solution

Answer

Answer: The slope of the given equation is -\frac{2}{3}, and the y-intercept is -4. To graph this equation, start by plotting the y-intercept (0, -4) on the graph, then use the slope to find a second point by moving down 2 units and right 3 units from the y-intercept. Plot this second point (3, -6) and draw a straight line through both points. This line represents the graph of the equation y = -\frac{2}{3}x - 4.

Step 1: Identify the slope and the y-intercept

To graph the equation, first identify the slope and the y-intercept. The equation is given in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. In this case, we have m = -\frac{2}{3} and b = -4.

Step 2: Plot the y-intercept

Start by plotting the y-intercept (0, -4) on the graph. This is the point where the line crosses the y-axis.

Step 3: Use the slope to plot a second point

The slope (-\frac{2}{3}) tells us the rise and run of the line. The rise is -2 (negative indicates a downward direction) and the run is 3 (positive indicates a rightward direction). To find a second point, move down 2 units (rise) and right 3 units (run) from the y-intercept (0, -4). This second point is (3, -6).

Step 4: Draw the line

Now that we have two points (0, -4) and (3, -6), we can plot them both on the graph. Then, draw a straight line through both points, making sure to extend the line past them to cover the entire graph.

Step 5: Verify your graph

Check that any point on the line will satisfy the equation y = -\frac{2}{3}x - 4 for any value of x. This verifies that your graph accurately represents the equation.

Key Concepts

Slope-intercept form

The slope-intercept form of a linear equation is a way of writing linear equations so that you can easily identify two key characteristics of the line: its slope and its y-intercept. This form is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) represents the y-intercept.

Here's why the slope-intercept form is useful for graphing:

• The formula clearly shows the slope \( m \), which tells you how steep the line is and in which direction it goes.
• The y-intercept \( b \) gives you the point where the line crosses the y-axis, making it easy to start plotting the graph.

When you see an equation in slope-intercept form, you can quickly sketch the line on a graph. First, plot the y-intercept, and then use the slope to determine the other points on the line. This method is efficient and minimizes calculation errors.

Slope

Slope is a concept that describes the direction and steepness of a line. In the slope-intercept form \( y = mx + b \), the \( m \) represents the slope. Slope is often referred to as "rise over run." Simply put, it indicates how much you go up or down (the rise) for every step you take across (the run).

If \( m = -\frac{2}{3} \) as in the example given, this means you'll move:

• 2 units down (because of the negative sign, indicating a downward direction)
• 3 units to the right (run, indicating the rightward direction)

The slope helps you understand the line's behavior:

• A positive slope means the line rises as it goes from left to right.
• A negative slope, like \(-\frac{2}{3}\), means the line falls as it progresses from left to right.
• A zero slope indicates a perfectly horizontal line, while an undefined slope represents a vertical line.

Thus, by knowing the slope, you gain insight into how the graph of the line will look, making it easier to draw and analyze.

Y-intercept

The y-intercept is a fundamental aspect of linear equations written in slope-intercept form. It is represented by \( b \) in the equation \( y = mx + b \). The y-intercept is the point where the line crosses the y-axis.

For the equation \( y = -\frac{2}{3}x - 4 \), the y-intercept is \(-4\). This means the line crosses the y-axis at the point \((0, -4)\). By plotting this point first, you establish a starting point for graphing.

Understanding the y-intercept allows you to:

• Quickly locate where your line begins on the y-axis.
• Serve as a reference point to apply the slope and find other points on the line.

Therefore, the y-intercept is crucial because it gives you an initial position from which to use the slope and extend the line across the graph. By mastering this concept, you simplify the task of graphing linear equations.