/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Graph the given equation. $$ ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 30

Graph the given equation. $$ y-\frac{1}{4} x=-2 $$

Short Answer

Expert verified
The given equation is \(y-\frac{1}{4}x=-2\). To graph it, rewrite it in slope-intercept form: \(y=\frac{1}{4}x-2\). Identify the slope (\(m=\frac{1}{4}\)) and the y-intercept (\(b=-2\)). Plot the y-intercept at (0, -2) and use the slope to find another point, (4, -1). Draw a line through the two points to graph the equation.

Step by step solution

01

Rewrite the equation in slope-intercept form

First, we want to rewrite the given equation \(y-\frac{1}{4}x=-2\) into the slope-intercept form, which is \(y = mx + b\). To do this, we simply need to isolate the y term by adding \(\frac{1}{4}x\) to both sides of the equation: \[ y = \frac{1}{4}x - 2 \]
02

Find the slope and the y-intercept

Now that we have the equation in slope-intercept form, we can identify the slope (m) and y-intercept (b) by comparing it with the general form \(y = mx + b\). We have: \[ m = \frac{1}{4},\ b = -2 \]
03

Plot the y-intercept

Next, we plot the y-intercept (-2) on the coordinate plane. To do this, we locate the point (0, -2) and put a point there.
04

Use the slope to find another point on the line

Now that we have the y-intercept, we can use our slope, \(\frac{1}{4}\), to locate another point on the line. Since the slope is the rise over run (螖y/螖x), we'll move from the y-intercept 1 unit up (rise) and 4 units to the right (run). This lands us at the point (4, -1). We can put a point at this new location.
05

Draw the line through the two points

Finally, we draw a line that passes through our y-intercept (0, -2) and our second point (4, -1). This line represents the graph of the equation \(y-\frac{1}{4}x=-2\).

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is one of the most popular ways to express a linear equation. It is written as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) denotes the y-intercept. This form is especially useful because it makes graphing a line straightforward: you can immediately see the slope and where the line crosses the y-axis.
  • The equation in slope-intercept form provides a clear picture of how the line behaves.
  • You can quickly identify the steepness (slope) and initial position (y-intercept) on the graph.
Converting an equation into this form usually involves simple algebraic operations, such as isolating \(y\) on one side. This clarity and ease of use make it a favorite when tackling graphing problems, like the one provided.
Plotting Points
Plotting points is the essential skill of placing points on a coordinate plane. To graph a linear equation, you typically need at least two points through which the line can be drawn. These points are often the y-intercept and another point found using the slope.
  • Start by plotting the y-intercept.
  • Use the slope to find and plot a second point.
Once you have two points, draw a line through them, extending it in both directions. Plotting correctly ensures your graph accurately represents the equation. It's a skill that becomes second nature with practice, making graphing much less daunting.
Slope of a Line
The slope of a line is a measure of its steepness and direction. In the slope-intercept form equation \(y = mx + b\), \(m\) is the slope. It tells us how much \(y\) changes for a change in \(x\).

Understanding Slope

  • A positive slope means the line ascends from left to right.
  • A negative slope means the line descends from left to right.
  • A slope of zero results in a horizontal line.
  • Undefined slope is a vertical line.
In our example, the slope is \(\frac{1}{4}\), indicating that for every 4 units moved right along the x-axis, the line rises 1 unit. Understanding slope is crucial for predicting how the line crosses the graph.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), the y-intercept is \(b\). It's essentially the value of \(y\) when \(x\) is zero. Knowing the y-intercept allows you to start graphing the line efficiently.
  • Identify the point where the line meets the y-axis. This is the y-intercept.
  • On a graph, this is represented by the point \( (0, b) \).
In the provided equation, the y-intercept is \(-2\). This tells us that the line will cross the y-axis at the point \( (0, -2) \). Starting with this knowledge makes plotting subsequent points easier and more accurate.

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Most popular questions from this chapter

If it costs Microsoft \(\$ 4,500\) to manufacture 8 Xbox 360 s and \(\$ 8,900\) to manufacture \(16,^{\dagger}\) obtain the corresponding linear cost function. What was the cost to manufacture each additional Xbox? Use the cost function to estimate the cost of manufacturing 50 Xboxes.

The Texas Bureau of Economic Geology published a study on the economic impact of using carbon dioxide enhanced oil recovery (EOR) technology to extract additional oil from fields that have reached the end of their conventional economic life. The following table gives the approximate number of jobs for the citizens of Texas that would be created at various levels of recovery. \(^{49}\) $$ \begin{array}{|r|c|c|c|c|} \hline \text { Percent Recovery (\%) } & 20 & 40 & 80 & 100 \\ \hline \text { Jobs Created (millions) } & 3 & 6 & 9 & 15 \\ \hline \end{array} $$ Find the regression line and use it to estimate the number of jobs that would be created at a recovery level of \(50 \%\).

The cost, in millions of dollars, of a 30-second television ad during the Super Bowl in the years 1990 to 2007 can be approximated by the following piecewise linear function \((t=0\) represents 1990\():{ }^{35}\) $$ C(t)=\left\\{\begin{array}{cc} 0.08 t+0.6 & \text { if } 0 \leq t<8 \\ 0.13 t+0.20 & \text { if } 8 \leq t \leq 17 \end{array}\right. $$ How fast and in what direction was the cost of an ad during the Super Bowl changing in \(2006 ?\)

Calculate the slope, if defined, of the straight line through the given pair of points. Try to do as many as you can without writing anything down except the answer. $$ (300,20.2) \text { and }(400,11.2) $$

Your friend April tells you that \(y=f(x)\) has the property that, whenever \(x\) is changed by \(\Delta x\), the corresponding change in \(y\) is \(\Delta y=-\Delta x .\) What can you tell her about \(f ?\)
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