Problem 5 - Algebra Form and Function

Question

Graph the equation. $$ y=3 x-6 $$

Step-by-Step Solution

Answer

Answer: The slope (m) of the equation $$y = 3x - 6$$ is 3, and the y-intercept (b) is -6. To graph the equation, plot the y-intercept at -6 on the y-axis, then use the slope to find additional points by moving up 3 units and to the right 1 unit from the y-intercept point. Connect the points to draw the line representing the equation.

Step 1: 1. Identify the slope and y-intercept

In the given equation, $$y = 3x - 6$$, the coefficient of x is 3, which represents the slope, and the constant term is -6, which represents the y-intercept. Therefore, the slope (m) is 3, and the y-intercept (b) is -6.

Step 2: 2. Plot the y-intercept

Since the y-intercept is -6, plot a point on the y-axis at -6. This point represents where the line intercepts the y-axis.

Step 3: 3. Use the slope to find additional points

The slope, 3, can be expressed as a fraction: $$\frac{3}{1}$$. This means that for every 3 units up on the y-axis, the line moves 1 unit to the right on the x-axis. Starting from the y-intercept point, move up 3 units and to the right 1 unit to find another point on the line. Repeat this process to find additional points to ensure accuracy.

Step 4: 4. Draw the line

Using the points found in step 3, draw a straight line connecting them. This line represents the equation $$y = 3x - 6$$.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express a linear equation. It makes graphing a line straightforward. The general form is given by the equation \(y = mx + b\). Here, \(m\) represents the slope of the line, while \(b\) represents the y-intercept. This form is particularly useful because it directly tells you how to start drawing the line.

• Starting point: The y-intercept, \(b\), is where the line crosses the y-axis.
• Direction: The slope, \(m\), indicates how steep the line is and its direction.

To effectively use this form, always identify the slope and y-intercept first, as it guides you on how to position and tilt the line on the graph.

Graphing Lines

Graphing lines using the slope-intercept form is a simple process. Start by plotting the y-intercept on the graph. This is your initial point and is located on the y-axis. For instance, in the equation \(y = 3x - 6\), the y-intercept is -6. So, you would place your first point at (0, -6).
Once you have your starting point, use the slope to determine the next points. The slope \(m\) can be written as a fraction \(\frac{rise}{run}\). Here, the rise indicates how much you move up or down, and the run indicates how much you move right or left.

• Positive slope: Move up and to the right.
• Negative slope: Move down and to the right.

By taking these steps, you can draw multiple points along the line, which helps ensure the accuracy of your graph.

Slope and Y-Intercept

Understanding both the slope and y-intercept is crucial for graphing linear equations accurately. The slope tells you how steep the line is. It is the change in the y-coordinate for each unit change in the x-coordinate. A slope of 3, as in our example \(y = 3x - 6\), means that for every 1 unit you move to the right, the line moves 3 units up.
The y-intercept is the point where the line crosses the y-axis. It's the value of \(y\) when \(x = 0\). For the equation \(y = 3x - 6\), the y-intercept is -6. This point is critical because it gives you a starting location from which to use the slope to determine the direction of the line.
Being able to correctly identify these elements not only aids in precise graphing but also enhances your understanding of the relationship between the algebraic equation and its graphical representation.