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Chapter 1: Problem 44

Find the equation and sketch the graph of the following lines. With slope \(\frac{1}{3}\) and \(y\) -intercept \((0,1)\)

Short Answer

Expert verified
The equation is \( y = \frac{1}{3}x + 1 \). Plot (0,1) and use slope to find more points.

Step by step solution

01

Identify the Given Information

The problem provides two key pieces of information: the slope and the y-intercept of the line. The slope is \(\frac{1}{3}\) and the y-intercept is the point \( (0,1) \).
02

Use the Slope-Intercept Form

The slope-intercept form of a line's equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
03

Substitute the Values Into the Equation

Substitute the slope value \(\frac{1}{3}\) for \( m \) and the y-intercept value 1 for \( b \) in the slope-intercept form: \[ y = \frac{1}{3}x + 1 \]
04

Graph the Equation

To graph the equation \ y = \frac{1}{3}x + 1 \, start by plotting the y-intercept \( (0,1) \). Next, use the slope to find another point. From (0,1), move up 1 unit and right 3 units to reach \ (3,2) \. Draw a line through these points.

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Key Concepts

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

Linear Equation
A linear equation represents a straight line on a graph. It is characterized by its slope (steepness) and y-intercept (the point at which the line crosses the y-axis). The most commonly used form to represent a linear equation is the slope-intercept form, given by \(y = mx + b\). Here, \(m\) stands for the slope, and \(b\) is the y-intercept.
y-intercept
The y-intercept of a line is the point where the line meets or crosses the y-axis. This point has an x-coordinate of zero. In the slope-intercept form of a linear equation, \(y = mx + b\), the value of \(b\) directly gives you the y-intercept. For instance, if a line equation is \(y = \frac{1}{3}x + 1\), the y-intercept is 1, meaning the line crosses the y-axis at the point (0,1).
Graphing Lines
Graphing a linear equation involves plotting points and drawing a straight line through them. To graph an equation like \(y = \frac{1}{3}x + 1\):
- Start by plotting the y-intercept, which is (0,1).
- Use the slope to find another point. Slope \(\frac{1}{3}\) means from the y-intercept, move up 1 unit (rise) and 3 units to the right (run). This gives a new point at (3,2).
- Plot the point (3,2) and draw a straight line through these points.
Remember, the slope \frac{1}{3}\ ensures the line inclines slightly upwards as it moves from left to right. Practice by graphing different lines using their unique slopes and y-intercepts to get a better understanding.

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Most popular questions from this chapter

The functions in Exercises 21-26 are defined for all \(x\) except for one value of \(x\). If possible, define \(f(x)\) at the exceptional point in a way that makes \(f(x)\) continuous for all \(x\). \(f(x)=\frac{x^{2}-7 x+10}{x-5}, x \neq 5\)

Compute the following limits. \(\lim _{x \rightarrow \infty} \frac{1}{x^{2}}\)

Compute the following limits. \(\lim _{x \rightarrow \infty} \frac{10 x+100}{x^{2}-30}\)

The functions in Exercises 21-26 are defined for all \(x\) except for one value of \(x\). If possible, define \(f(x)\) at the exceptional point in a way that makes \(f(x)\) continuous for all \(x\). \(f(x)=\frac{x^{2}+x-12}{x+4}, x \neq-4\)

Compute the difference quotient $$ \frac{f(x+h)-f(x)}{h} . $$ Simplify your answer as much as possible. \(f(x)=x^{3}\) [Hint: \((a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}\).]
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