Problem 39 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$y=2 x-8$$

Step-by-Step Solution

Answer

The graph is a line passing through the points (0, -8) and (4, 0).

Step 1: Understand the Equation

The equation given is a linear equation in the slope-intercept form, which is given by: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Identify Slope and Y-Intercept

From the equation \(y = 2x - 8\), identify that the slope \(m\) is 2 and the y-intercept \(b\) is -8.

Step 3: Find the Y-Intercept

The y-intercept is the point where the graph crosses the y-axis. Set \(x = 0\) in the equation and solve for \(y\): \(y = 2(0) - 8 = -8\). So, the y-intercept is (0, -8).

Step 4: Find the X-Intercept

The x-intercept is the point where the graph crosses the x-axis. Set \(y = 0\) in the equation and solve for \(x\): \(0 = 2x - 8\) \(2x = 8\) \(x = 4\). So, the x-intercept is (4, 0).

Step 5: Plot the Intercepts

Plot the intercepts (0, -8) and (4, 0) on the coordinate plane.

Step 6: Draw the Line

Draw a straight line through the points (0, -8) and (4, 0) to sketch the graph of the equation.

Key Concepts

graphing linear equations

Graphing a linear equation involves drawing its line on a coordinate plane. The given equation,

y = 2x - 8  

involves several steps to transform this algebraic form into a visual representation.
First, identify important points like intercepts. These anchor the graph and make it easier to draw the line accurately. Next, plot these points on your coordinate plane.
In this example, we found the intercepts:

• The y-intercept: (0, -8)
• The x-intercept: (4, 0)

With these points plotted, use a straight edge to connect them. The resulting line represents all possible solutions to the equation. The line will extend infinitely in both directions.

slope-intercept form

Understanding the slope-intercept form is crucial when dealing with linear equations. This form is written as:

y = mx + b 

Here, 'm' is the slope and 'b' is the y-intercept.
In our equation,

y = 2x - 8  

The slope (m) is 2, which indicates that for every unit increase in x, y increases by 2 units. The y-intercept (b) is -8, meaning the graph crosses the y-axis at (0, -8).
Knowing these values helps in quickly sketching the graph and understanding the line's steepness and starting point on the y-axis.

finding intercepts

Intercepts are where the graph crosses the axes. Understanding these points provides an easy start for graphing the equation.

• Finding the Y-Intercept: Set x to 0 and solve for y. In our equation: y = 2(0) - 8 = -8. Hence, the y-intercept is (0, -8).
• Finding the X-Intercept: Set y to 0 and solve for x. For our equation: 0 = 2x - 8. Solving for x: 2x = 8, x = 4. Thus, the x-intercept is (4, 0).

Once you have these intercepts, you can easily plot them and draw the line passing through them to graph the equation.