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Chapter 6: Problem 4

A linear function is given. Determine the \(x\) -intercept and \(y-\) intercept, and then graph the linear function. If the function does not have an \(x\) -intercept, then say so. $$f(x)=12+3 x$$

Short Answer

Expert verified
x-intercept: (-4,0), y-intercept: (0,12).

Step by step solution

01

Identify the Linear Function

The given linear function is \( f(x) = 12 + 3x \). This function is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Determine the y-intercept

From the function \( f(x) = 12 + 3x \), it is evident that the y-intercept \( b \) is \( 12 \). This is the point where the line crosses the y-axis, so when \( x = 0 \), \( y = 12 \). Thus, the y-intercept is \( (0, 12) \).
03

Determine the x-intercept

To find the x-intercept, set \( f(x) = 0 \) and solve for \( x \):\[ 0 = 12 + 3x \]\[ -12 = 3x \]\[ x = -4 \]So, the x-intercept is \( (-4, 0) \). This is the point where the line crosses the x-axis.
04

Graph the Linear Function

To graph the linear function, plot the y-intercept \( (0,12) \) and the x-intercept \( (-4,0) \) on the coordinate plane. Draw a straight line through these two points. This line represents the linear equation \( f(x) = 12 + 3x \).

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Key Concepts

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

Understanding X-Intercepts
The x-intercept of a linear function is the point where the graph of the function crosses the x-axis. At this point, the value of the function, or y, is zero. To find the x-intercept, you set the function equal to zero and solve for x. This essentially means we're looking for a point on the line where there is no vertical shift from the x-axis.

For our function \( f(x) = 12 + 3x \), we solve \( 0 = 12 + 3x \) to find the x-intercept.
By rearranging, we get:
\(-12 = 3x\)
\(x = -4\)
So, the x-intercept is at the point \((-4, 0)\).

Knowing the x-intercept can help us understand where a graph crosses or meets the horizontal axis, revealing key changes in trends or behaviors in applied contexts.
Exploring Y-Intercepts
The y-intercept of a linear function is the point where the graph crosses the y-axis. At this location, the value of x is always zero. The y-intercept is available directly from the slope-intercept form of the equation, given as \( y = mx + b \), where \( b \) is the y-intercept.

In our example with \( f(x) = 12 + 3x \), the y-intercept \( b \) is 12. Therefore, the y-intercept is the point \((0, 12)\).
This means that when there is no change in x, the function starts at 12 on the y-axis.
This piece of information serves as a crucial starting point in graphing the line and understanding the behavior of the function when x is zero.
Getting to Know Slope-Intercept Form
The slope-intercept form of a linear function provides a clear and concise way to describe the equation of a line. This form is expressed as \( y = mx + b \), where \( m \) represents the slope, and \( b \) represents the y-intercept.

Knowing this format allows us to easily interpret and graph linear equations. In our linear function \( f(x) = 12 + 3x \), the slope \( m \) is 3, indicating that for every unit increase in x, the function value, or y, increases by 3. The y-intercept \( b \) is 12, which as previously discussed, is where the line crosses the y-axis when x is 0.

This form is very user-friendly for both calculating and graphing lines, enabling quick identification of important characteristics like slope and intercepts.

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Most popular questions from this chapter

Consider the following scenario: Australia's Productivity Commission wants to exclude business software from patents under the country's laws. The number of patent applications in Australia from 2005 to 2013 can be expressed by the function $$F(x)=1214-32.7 x, \quad x=5,6, \ldots, 13$$ where \(x\) represents the number of years since 2000 and \(F(x)\) represents the number of patents issued that year. Use this model to answer the following questions. How many patents were issued in \(2005 ?\)

Consider the following scenario: A recent point for discussion has been an increase in the national minimum wage. Joe Roman of the Greater Cleveland Partnership claims that an increase in the minimum wage to \(\$ 15\) per hour in Cleveland would be the "most aggressive minimum wage increase in the country". The annual monthly minimum-wage earnings in the United States from 2007 to 2015 can be modeled by the function $$f(x)=419.9+41.2 x, \quad x=7,8, \ldots, 15$$ where \(x\) represents the number of years since 2000 and \(f(x)\) represents the monthly minimum-wage earning in dollars. Use this function to answer the following questions. Overall, did the monthly minimum wage increase or decrease from 2007 to \(2015 ?\)

Find the area under \(f(x)\) on the indicated interval. Round the area to two decimal places as necessary. $$f(x)=\left\\{\begin{array}{cl} 1.1 x, & 0 \leq x \leq 11 \\ 0, & \text { otherwise } \end{array} \text { on the interval } 5 \leq x \leq 10\right.$$

Find the area under \(f(x)\) on the indicated interval. Round the area to two decimal places as necessary. $$f(x)=\left\\{\begin{array}{cl} 2.5 x, & 0 \leq x \leq 5 \\ 0, & \text { otherwise } \end{array} \text { on the interval } 2 \leq x \leq 5\right.$$

Determine the area under each constant function on the indicated interval. Then graph the result. $$F(x)=\left\\{\begin{array}{ll} 0.8, & 0 \leq x \leq 11 \\ 0, & \text { otherwise } \end{array} \text { on the interval } 7.4 \leq x \leq 10.1\right.$$
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