Problem 70 - Algebra 1: Concepts and Skills

Question

Use a table of values to graph the equation. Label the x-intercept and the y-intercept. \(y=4 x-4\)

Step-by-Step Solution

Answer

The graph of the equation \(y = 4x - 4\) is a straight line passing through points (-1,-8), (0,-4), and (1,0). The y-intercept is (0,-4) and the x-intercept is (1,0).

Step 1: Create the Table Values

Start by selecting a variety of values for x and then substitute these values into the given equation to find the corresponding y values. Choose both negative and positive values for x. For instance, select \(x=-1\), \(x=0\) and \(x=1\). By substituting these values into the equation, we get \(y=4(-1)-4=-8\), \(y=4(0)-4=-4\), and \(y=4(1)-4=0\) respectively. Hence the table will have values (-1,-8), (0,-4) and (1,0).

Step 2: Plotting the Values on the Graph

Graph the equation using the values from the table. Place the x values on the horizontal axis and the y values on the vertical axis. Plot the points (-1,-8), (0,-4), and (1,0) on the graph.

Step 3: Drawing the Graph Line and Marking the Intercepts

Draw a line through the points plotted on the graph. The x-intercept is the point where the line crosses the x-axis, while the y-intercept is where it crosses the y-axis. In this case, the y-intercept is (0,-4) and the x-intercept is (1,0). Label these points on the graph.

Key Concepts

Understanding the x-intercept

The x-intercept of a graph is an important concept when studying linear equations. It is the point where the graph intersects the x-axis. This means that at the x-intercept, the value of y is zero. To find the x-intercept from an equation, you set the y-value to 0 and solve for x.
For the equation given in this problem, which is \(y = 4x - 4\), we find the x-intercept by setting \(y = 0\):

• Substitute 0 for y: \(0 = 4x - 4\)
• Solve for x: \(4x = 4\)
• Divide by 4: \(x = 1\)

Therefore, the x-intercept is at the point (1,0). This is the point where the line crosses the x-axis on the graph.

Identifying the y-intercept

The y-intercept is another key concept when dealing with linear equations and graphing. It represents the point where the line crosses the y-axis. At the y-intercept, the value of x is zero.
To find the y-intercept, we set the value of x to 0 in the equation and solve for y.
For the equation \(y = 4x - 4\), find the y-intercept as follows:

• Substitute 0 for x: \(y = 4(0) - 4\)
• Calculate y: \(y = -4\)

So, the y-intercept of the line is at the point (0,-4). This intercept is crucial as it is often used to initiate drawing the line on a coordinate plane.

Creating a table of values

Creating a table of values is a practical method for graphing equations, as it helps visualize how different x-values relate to their corresponding y-values.
To create a table of values, you select various values for x, plug them into the equation, and solve for y. It helps to choose a mix of negative and positive values to see how the line behaves on either side of the axes.
For our example \(y = 4x - 4\) in the step-by-step solution:

• Choose \(x = -1\), \(x = 0\), and \(x = 1\)
• Calculate y for each x:
  • \(x = -1\), then \(y = 4(-1) - 4 = -8\)
  • \(x = 0\), then \(y = 4(0) - 4 = -4\)
  • \(x = 1\), then \(y = 4(1) - 4 = 0\)

These values make up the pairs (-1, -8), (0, -4), and (1, 0) and can be plotted to ensure the graph is accurate. This table not only aids in drawing the line but also in identifying the intercepts as shown above.