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Chapter 1: Problem 29

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. \(y=\frac{x+2}{3}\)

Short Answer

Expert verified
The slope is \(\frac{1}{3}\) and the \(y\)-intercept is \((0, \frac{2}{3})\).

Step by step solution

01

Identify Equation Form

The equation given is \(y = \frac{x+2}{3}\). We need to reformat it into the slope-intercept form \(y = mx + b\). To do this, we will simplify \(\frac{x+2}{3}\).
02

Simplify the Equation

Simplify \(\frac{x+2}{3}\) by distributing the division across the terms: \(y = \frac{1}{3}x + \frac{2}{3}\). Now it matches the slope-intercept form with \(m = \frac{1}{3}\) and \(b = \frac{2}{3}\).
03

Identify the Slope and Y-Intercept

In the equation \(y = \frac{1}{3}x + \frac{2}{3}\), the slope \(m\) is \(\frac{1}{3}\) and the \(y\)-intercept \(b\) is \(\frac{2}{3}\). Thus, the \(y\)-intercept point is \((0, \frac{2}{3})\).
04

Plot the Graph

Start by plotting the \(y\)-intercept \((0, \frac{2}{3})\) on the graph. From this point, use the slope \(\frac{1}{3}\) to find another point. The slope suggests that for a run of 3 units in the positive \(x\)-direction (right), the rise is 1 unit up in the \(y\)-direction. Plot this second point and draw a straight line through the points.

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Key Concepts

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

Slope-Intercept Form
Understanding the slope-intercept form of a linear equation is crucial for analyzing and graphing linear functions. The slope-intercept form is given by the equation \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) represents the y-intercept. This form makes it easy to identify both the rate of change of the line and the point at which it crosses the y-axis.

When an equation is in this form:
  • The coefficient of \(x\) (\(m\)) tells us how steep the line is. A larger value indicates a steeper slope.
  • The constant \(b\) shows where the line touches the y-axis. This is the y-intercept.
By converting equations into the slope-intercept form, recognizing these components becomes straightforward, aiding in the process of graphing linear functions.
Graphing Linear Functions
Graphing linear functions is about effectively translating the formula of a line into a visual depiction on a coordinate plane. To graph a linear equation, first ensure it's in the slope-intercept form, \(y = mx + b\).

Follow these steps:
  • Start by plotting the y-intercept \(b\) on the y-axis. This point is \((0, b)\), where the line crosses the y-axis.
  • From this point, use the slope \(m\) to determine the direction and steepness of the line. A positive slope means the line tilts upwards as it moves from left to right, while a negative slope means it goes down.
  • Move from the y-intercept according to the slope's rise over run. For instance, if \(m = \frac{1}{3}\), it indicates a rise of 1 unit for every 3 units moved to the right.
  • After plotting the necessary points, draw a straight line through them to complete the graph.
Graphing linear functions provides a clear visual understanding of the relationship defined by the equation.
Slope and Y-Intercept
In any linear equation, the slope and y-intercept encapsulate significant information about the line. The slope \(m\) represents the rate of change, or how much \(y\) increases or decreases as \(x\) changes. A greater absolute value of the slope corresponds to a steeper incline or decline of the line.

  • The slope signifies direction:
    • Positive slopes mean the line rises as it goes from left to right.
    • Negative slopes mean the line descends.
    • A zero slope indicates a horizontal line, which means the function has constant output.
  • The y-intercept \(b\) is the starting point of the function on the y-axis when \(x = 0\). This is where the function intersects the y-axis.
  • Knowing both the slope and the y-intercept gives complete insight into the behavior of the linear function.
Understanding these concepts simplifies analyzing linear relationships and aids in predicting the behavior of functions across different values of \(x\).

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Most popular questions from this chapter

BUSINESS: Break-Even Points and Maximum Profit A sporting goods store finds that if it sells \(x\) exercise machines per day, its costs will be \(C(x)=100 x+3200\) and its revenue will be \(R(x)=-2 x^{2}+300 x\) (both in dollars). a. Find the store's break-even points. b. Find the number of sales that will maximize profit, and the maximum profit.

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=\sqrt{x} $$

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=3 x^{2}-5 x+2 $$

True or False: \(\infty\) is the largest number.

ATHLETICS: Juggling If you toss a ball \(h\) feet straight up, it will return to your hand after \(T(h)=0.5 \sqrt{h}\) seconds. This leads to the juggler's dilemma: Juggling more balls means tossing them higher. However, the square root in the above formula means that tossing them twice as high does not gain twice as much time, but only \(\sqrt{2} \approx 1.4\) times as much time. Because of this, there is a limit to the number of balls that a person can juggle, which seems to be about ten. Use this formula to find: a. How long will a ball spend in the air if it is tossed to a height of 4 feet? 8 feet? b. How high must it be tossed to spend 2 seconds in the air? 3 seconds in the air?
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