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Chapter 5: Problem 30

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

Short Answer

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

Step by step solution

01

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.
02

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).
03

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).
04

Plot the intercepts

On a graph, plot the points (0, 12) for the y-intercept and (-3, 0) for the x-intercept.
05

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.
06

Label the intercepts

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Solve each of the following problems algebraically. Be sure to label what the variable represents. The width of a rectangle is 4 less than \(\frac{2}{3}\) its length. If the perimeter of the rectangle is 3 times the length, find its dimensions.

Sets of values are given for variables having a linear relationship. In each case, write the slope-intercept form for the equation of the line corresponding to the given set of values and answer the accompanying question. $$\begin{array}{|l|c|c|} \hline x \text { (Number of hours spent studying) } & 2 & 3 \\ \hline y \text { (Grade on math exam) } & 75 & 82 \\ \hline \end{array}$$ What would the grade be if a student studies for 5 hours?

Sketch the graph of the line satisfying the given conditions. Passing through \((-1,-5)\) with 0 slope

Sketch the graph of the line satisfying the given conditions. Passing through \((-1,0)\) with slope \(-4\)

How could you use the idea of slope to show that the three points \((-1,-2)\) \((2,0),\) and \((5,2)\) all lie on a straight line?
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