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Chapter 2: Problem 23

Graph the function. Find the slope, \(y\) -intercept and \(x\) -intercept, if any exist. \(f(x)=3\)

Short Answer

Expert verified
It's a horizontal line at \( y = 3 \), slope is 0, y-intercept is \( (0, 3) \), no x-intercept.

Step by step solution

01

Understand the Function

The function given is a constant function, \( f(x) = 3 \). This means that for any value of \( x \), \( f(x) \, = \, y \, = \, 3 \). It is a horizontal line.
02

Graph the Function

To graph \( f(x) = 3 \), draw a horizontal line that passes through the point where \( y = 3 \) on the y-axis. This line will be parallel to the x-axis.
03

Identify the Slope

Since the line is horizontal, the slope is \( 0 \). The slope of a line \( m \) is defined by \( m = \frac{\Delta y}{\Delta x} \). In a horizontal line, there is no change in \( y \) as \( x \) changes, thus \( \Delta y = 0 \) and \( m = 0 \).
04

Find the Y-intercept

The y-intercept is the point where the line crosses the y-axis. For \( f(x) = 3 \), the line crosses the y-axis at the point \( y = 3 \). Hence, the y-intercept is \( (0, 3) \).
05

Find the X-intercept

The x-intercept is the point where the line crosses the x-axis. Since the line \( f(x) = 3 \) is parallel to the x-axis and does not touch or cross it, there is no x-intercept.

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Key Concepts

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

Understanding the Constant Function
When we talk about a constant function, we're dealing with a function where the output value remains the same for any input value. For instance, in the function \( f(x) = 3 \), no matter what \( x \) is, the value of \( f(x) \) or \( y \) will always be 3. This creates a horizontal line when graphed.
This kind of function can be helpful for visualizing how some variables do not change with others. In this case, the constant function is simply a flat, straight line parallel to the x-axis at the height of 3 on the y-axis.
  • Graph: Horizontal line
  • Equation form: \( y = c \) where \( c \) is a constant
A constant function like this highlights the simplicity of horizontal lines and helps when learning other types of functions.
Understanding the Slope of a Line
The slope of a line tells us how steep the line is. It's a measure of the line's angle compared to the axes. In mathematical terms, the slope \( m \) is calculated using the formula \( m = \frac{\Delta y}{\Delta x} \), which means 'change in y over change in x'.
For our constant function, \( f(x) = 3 \), this slope is 0, because a horizontal line does not rise or fall as it moves from left to right. There is no change in the y-value, resulting in \( \Delta y = 0 \).
  • No vertical change in the line
  • Slope (\( m \)) = 0
This zero slope confirms we're dealing with a flat, even line across the graph.
Locating the Y-intercept
The y-intercept is a key feature to find when graphing a line. This is the point where the line intersects the y-axis, essentially where the line hits the vertical axis.
For the function \( f(x) = 3 \), the line directly crosses the y-axis at the point (0, 3). This happens because for any line in this constant position, the y-coordinate remains constant while \( x \) is zero at the y-axis crossover.
  • Y-intercept point: \( (0,3) \)
Understanding the y-intercept helps you quickly determine where the line will position itself on the graph.
Examining the X-intercept
The x-intercept is another important concept in graphing that tells us where the line crosses the x-axis. Unlike the y-intercept, the x-intercept of a horizontal line like \( f(x) = 3 \) does not exist. This is because the line remains parallel to the x-axis.
Since it is a consistent line that never touches the x-axis, there is no point on the line where \( y \) equals zero.
  • No x-intercept exists in this scenario
This knowledge allows you to anticipate how horizontal lines behave on a graph, especially when examining intercepts.

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Most popular questions from this chapter

On New Year's Day, I (Jeff, again) started weighing myself every morning in order to have an interesting data set for this section of the book. (Discuss with your classmates if that makes me a nerd or a geek. Also, the professionals in the field of weight management strongly discourage weighing yourself every day. When you focus on the number and not your overall health, you tend to lose sight of your objectives. I was making a noble sacrifice for science, but you should not try this at home.) The whole chart would be too big to put into the book neatly, so I've decided to give only a small portion of the data to you. This then becomes a Civics lesson in honesty, as you shall soon see. There are two charts given below. One has my weight for the first eight Thursdays of the year (January 1, 2009 was a Thursday and we'll count it as Day \(1 .\) ) and the other has my weight for the first 10 Saturdays of the year. $$\begin{array}{|l|r|r|r|r|r|r|r|r|}\hline \begin{array}{l}\text { Day # } \\ \text { (Thursday) }\end{array} & 1 & 8 & 15 & 22 & 29 & 36 & 43 & 50 \\\\\hline \begin{array}{l}\text { My weight } \\\\\text { in pounds }\end{array} & 238.2 & 237.0 & 235.6 & 234.4 & 233.0 & 233.8 & 232.8 & 232.0\\\\\hline\end{array}$$ $$\begin{array}{|l|r|r|r|r|r|r|r|r|r|r|}\hline \begin{array}{l}\text { Day # } \\\\\text { (Saturday) } \end{array} & 3 & 10 & 17 & 24 & 31 & 38 & 45 & 52 & 59 & 66 \\\\\hline \begin{array}{l}\text { My weight } \\\\\text { in pounds }\end{array} & 238.4 & 235.8 & 235.0 & 234.2 & 236.2 & 236.2 & 235.2 & 233.2 & 236.8 & 238.2 \\\\\hline\end{array}$$ (a) Find the least squares line for the Thursday data and comment on its goodness of fit. (b) Find the least squares line for the Saturday data and comment on its goodness of fit. (c) Use Quadratic Regression to find a parabola which models the Saturday data and comment on its goodness of fit. (d) Compare and contrast the predictions the three models make for my weight on January 1, 2010 (Day #366). Can any of these models be used to make a prediction of my weight 20 years from now? Explain your answer. (e) Why is this a Civics lesson in honesty? Well, compare the two linear models you obtained above. One was a good fit and the other was not, yet both came from careful selections of real data. In presenting the tables to you, I have not lied about my weight, nor have you used any bad math to falsify the predictions. The word we're looking for here is 'disingenuous'. Look it up and then discuss the implications this type of data manipulation could have in a larger, more complex, politically motivated setting. (Even Obi-Wan presented the truth to Luke only "from a certain point of view.")

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