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Chapter 3: Problem 36

Graph each linear equation. See Examples 2 through 6. $$ y=\frac{1}{4} x+3 $$

Short Answer

Expert verified
Plot \((0, 3)\) and \((1, 3.25)\) and draw the line through them.

Step by step solution

01

Identify the Slope and Y-Intercept

The equation of a line in slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In the given equation \( y = \frac{1}{4} x + 3 \), the slope \( m \) is \( \frac{1}{4} \) and the y-intercept \( b \) is \( 3 \).
02

Plot the Y-Intercept

Start by plotting the y-intercept on the graph. The y-intercept \( b \) is \( 3 \), so plot the point \((0, 3)\) on the y-axis.
03

Use the Slope to Plot Another Point

The slope \( \frac{1}{4} \) means that for every 1 unit you move right on the x-axis, you move up \( \frac{1}{4} \) unit on the y-axis. From the y-intercept \((0, 3)\), move 1 unit to the right (to \( x = 1 \)) and \( \frac{1}{4} \) unit up to plot the point \((1, 3.25)\).
04

Draw the Line

With the y-intercept point \((0, 3)\) and the point \((1, 3.25)\), use a ruler to draw the straight line that extends through these points across the graph. This represents the graph of the equation \( y = \frac{1}{4}x + 3 \).

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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 of a linear equation is one of the most common ways to express a straight line. This form is written as \( y = mx + b \). It provides useful information like the slope and the y-intercept directly from the equation.
  • \( y \) is the dependent variable, which you find values for.
  • \( x \) is the independent variable, you have control over this value.
  • \( m \) is the slope of the line, indicating how steep the line is.
  • \( b \) is the y-intercept, showing where the line crosses the y-axis.
This format is particularly useful because it allows you to quickly sketch the graph of the line by starting at the y-intercept and using the slope. Recognizing an equation in slope-intercept form is important for graphing and understanding the behavior of linear relationships.
Slope
The slope of a line is a measure of its steepness. In the slope-intercept form equation \( y = mx + b \), \( m \) represents the slope. Here's how it works:
  • A positive slope means the line is rising as it moves from left to right.
  • A negative slope means the line is falling as it moves from left to right.
  • If the slope is zero, the line is horizontal.
  • If the slope is undefined, the line is vertical.
For the equation \( y = \frac{1}{4}x + 3 \), the slope is \( \frac{1}{4} \). This means for every increase of 1 unit on the x-axis, there is an increase of \( \frac{1}{4} \) unit on the y-axis. The smaller the slope, the less steep the line.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. This is easily identified in the slope-intercept form \( y = mx + b \). Here, \( b \) is the y-intercept.
For example, in the equation \( y = \frac{1}{4}x + 3 \), the y-intercept is 3. This means the point \((0,3)\) is where the line crosses the y-axis.
The y-intercept provides a starting point for graphing the line. Once you plot this point, you can use the slope to determine the line's direction and steepness. This integral starting point helps form a complete picture of the linear equation's graph on a coordinate plane.

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Most popular questions from this chapter

Find the slope of the line that is (a) parallel and (b) perpendicular to the line through each pair of points. \((6,-2)\) and \((1,4)\)

Find the slope of the line through the given points. \((2.3,0.2)\) and \((7.9,5.1)\)

The average price of an acre of U.S. farmland was \(\$ 1132\) in \(2001 .\) In \(2006,\) the price of an acre rose to approximately \(\$ 1657 .\) (Source: National Agricultural Statistics Service) a. Write two ordered pairs of the form (year, price of acre) b. Find the slope of the line through the two points. c. Write a sentence explaining the meaning of the slope as a rate of change.

Example $$\text { If } f(x)=x^{2}+2 x+1, \text { find } f(\pi)$$ Solution: $$\begin{aligned} &f(x)=x^{2}+2 x+1\\\ &f(\pi)=\pi^{2}+2 \pi+1 \end{aligned}$$ Given the following functions, find the indicated values. $$f(x)=x^{2}-12$$ a. \(f(12)\) b. \(f(a)\)

See Examples 1 through 7 . Find an equation of each line described. Write each equation in slope-intercept form (solved for \(y\) ), when possible. Through \((6,7),\) parallel to the \(x\) -axis
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