Problem 30 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the given equation. Label the intercepts. $$y=4 x+12$$

Step-by-Step Solution

Answer

Intersects at (0,12) on y-axis and (-3,0) on x-axis.

Step 1: Identify the equation type

The given equation, \(y = 4x + 12\), is a linear equation in slope-intercept form where \(y = mx + b\). Here, the slope \(m\) is 4 and the y-intercept \(b\) is 12.

Step 2: Find the y-intercept

The y-intercept is the point where the line crosses the y-axis. For \(y = 4x + 12\), the y-intercept (where \(x = 0\)) is \(y = 12\). So, the coordinate of the y-intercept is (0, 12).

Step 3: Find the x-intercept

The x-intercept is the point where the line crosses the x-axis. This happens when \(y = 0\). Setting \(y = 0\) in the equation \(0 = 4x + 12\) and solving for \(x\), we get: \[ 0 = 4x + 12 \] \[ 4x = -12 \] \[ x = -3 \]So, the coordinate of the x-intercept is (-3, 0).

Step 4: Plot the intercepts

On a graph, plot the points (0, 12) for the y-intercept and (-3, 0) for the x-intercept.

Step 5: Draw the line

Draw a straight line through the points (0, 12) and (-3, 0) to represent the equation \(y = 4x + 12\). This line is the graph of the given equation.

Step 6: Label the intercepts

Clearly label the intercepts on the graph: (0, 12) is the y-intercept and (-3, 0) is the x-intercept.

Key Concepts

plotting points

Plotting points is a straightforward yet crucial step in graphing linear equations. Start by plotting the y-intercept, as it gives a clear starting point on the graph. In our example, the y-intercept is \((0, 12)\). Next, plot the x-intercept which we found to be \((-3, 0)\). Plotting these points gives you two fixed points through which the line must pass. Draw a straight line through these points to represent the equation. Make sure to label the intercepts clearly so you can easily identify these key points on your graph. This method ensures that anyone looking at your graph can immediately understand the relationship between the variables.