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Chapter 3: Problem 30

Graph each equation by hand. $$ y=\frac{1}{4} x $$

Short Answer

Expert verified
Plot points (0,0) and (4,1) then draw the line through them.

Step by step solution

01

Identify the Slope

The given equation is in the form of the slope-intercept form, which is: \(y = mx + b\). Here, the slope \(m\) is \(\frac{1}{4}\) and the y-intercept \(b\) is \(0\).
02

Plot the Y-Intercept

Since the y-intercept is \(0\), plot the point \((0, 0)\) on the graph.
03

Use the Slope to Find Another Point

Starting from the y-intercept \((0,0)\), use the slope \(\frac{1}{4}\). This means for every 4 units you move right, you move 1 unit up. From the point \((0,0)\), move 4 units to the right to \(x=4\) and 1 unit up to \(y=1\), giving the point \((4,1)\). Plot this point \((4,1)\).
04

Draw the Line

Draw a straight line through the points \((0,0)\) and \((4,1)\). This is the graph of the equation \(y = \frac{1}{4} x\).

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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 a common way to write the equation of a line. It looks like this:

\(y = mx + b\)

Where:
  • \(m\) represents the slope of the line.
  • \(b\) represents the y-intercept of the line.
This form is very useful because it gives us both the slope and the y-intercept right away. It is easy to graph a line when it is written in slope-intercept form. First, we can identify the y-intercept, then use the slope to find other points on the line.
Slope
The slope of a line describes how steep the line is. It shows the rate of change between the y-values and x-values. In the slope-intercept form \(y = mx + b\), the slope is represented by \(m\).

A positive slope means that as \(x\) increases, \(y\) also increases. A negative slope means that as \(x\) increases, \(y\) decreases.

In our example, the slope \(m\) is \(\frac{1}{4}\). This means for every 4 units you move to the right along the x-axis, you move 1 unit up on the y-axis.

To summarize:
  • Slope = Rise/Run
  • Positive slope: Line goes upwards
  • Negative slope: Line goes downwards
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 represented by \(b\).

To find it, just look at the value of \(b\). This gives the point \((0, b)\). For example, if \(b = 3\), then the line crosses the y-axis at \((0, 3)\).

In our exercise, the y-intercept \(b\) is \(0\). This means the line crosses the y-axis at the origin \((0, 0)\).

Important points:
  • The y-intercept tells us where the line starts on the y-axis.
  • It's very easy to plot this point on the graph.
Plotting Points
To graph a line, we need to plot points. Here's how to do it step by step:

  • Start by plotting the y-intercept. This is our first point and it usually makes graphing easier.
  • Next, use the slope to find additional points. In our example, start from the y-intercept \((0, 0)\), move 4 units to the right and 1 unit up to get to the point \((4, 1)\).
  • Finally, draw a line through the points you have plotted. This will be your graph.


Practice makes perfect. Try graphing different lines using these steps.

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Most popular questions from this chapter

In \(1993,\) the life expectancy at birth of females was 78.8 years. In \(2006,\) it was 80.2 years. Let \(E\) represent life expectancy and \(t\) the number of years since 1990 Source: National Vital Statistics Reports a) Find a linear equation that fits the data. b) Predict the life expectancy of females in \(2010 .\)

Find the indicated function values for each function. $$f(x)=\left\\{\begin{array}{ll}{x^{2}-10,} & {\text { if } x<-10} \\\\{x^{2},} & {\text { if }-10 \leq x \leq 10} \\\\{x^{2}+10,} & {\text { if } x>10}\end{array}\right.$$ a) \(f(-10)\) b) \(f(10)\) c) \(f(11)\)

Water freezes at \(32^{\circ}\) Fahrenheit and at \(0^{\circ}\) Celsius. Water boils at \(212^{\circ} \mathrm{F}\) and at \(100^{\circ} \mathrm{C} .\) What Celsius temperature corresponds to a room temperature of \(70^{\circ} \mathrm{F} ?\)

Write a slope-intercept equation of the line whose graph is described. Parallel to the graph of \(2 x-y=1 ; y\) -intercept \((0,-3)\)

In Exercises \(101-108\), each model is of the form \(f(x)=m x+b .\) In each case, determine what \(m\) and \(b\) signify. Life Expectancy of American Women. The life expectancy of American women \(t\) years after 1970 is given by \(A(t)=\frac{1}{7} t+75.5\)
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