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Chapter 3: Problem 42

Find the slope of the graph of \(f(x)=8 .\) If the slope is undefined, state this. [ 2.4]

Short Answer

Expert verified
The slope is 0.

Step by step solution

01

- Identify the type of function

The function given is \( f(x) = 8 \), which represents a constant function. This means that for any value of \( x \), the output \( f(x) \) remains constant at 8.
02

- Understand the graph of a constant function

The graph of a constant function is a horizontal line. For \( f(x) = 8 \), the line is horizontal and intersects the y-axis at 8.
03

- Determine the slope of a horizontal line

The slope of a line measures its steepness. Since a horizontal line does not rise or fall as it moves from left to right, it has a slope of 0.

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Key Concepts

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

slope of a line
The concept of the slope of a line is fundamental in understanding how graphs behave. The slope measures how steep a line is. Formally, it is the ratio of the change in the y-values to the change in the x-values between two points on the line. This is often expressed as: \[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \] where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on the line.
For example, if you have points (3, 4) and (7, 8) on the line, the slope \( m \) is calculated as follows:
  • Change in y-values (\( y_2 - y_1 \)): 8 - 4 = 4
  • Change in x-values (\( x_2 - x_1 \)): 7 - 3 = 4
  • Slope (\( m \)): \( \frac{4}{4} = 1 \)
This indicates a line that rises one unit vertically for each unit it moves horizontally.
horizontal line
Horizontal lines are special types of lines on a graph. They are characterized by having a slope of zero because they do not rise or fall as they move from left to right. In other words, the y-value remains the same no matter what the x-value is.
For instance, the equation \( y = 8 \) describes a horizontal line where all y-values are equal to 8, no matter what x-value you choose. To find the slope of this line:
  • Select any two points, say (1, 8) and (2, 8).
  • Calculate the difference in y-values: \( 8 - 8 = 0 \)
  • Calculate the difference in x-values: \( 2 - 1 = 1 \)
  • The slope \( m \) is \( \frac{0}{1} = 0 \)
This horizontal line does not change vertically, confirming that the slope is indeed zero.
graph of a function
The graph of a function visually represents the relationship between the input (x-values) and the output (y-values) of a function. Each point on the graph corresponds to a pair \( (x, f(x)) \).For the function \( f(x) = 8 \), the output is always 8, regardless of the x-value. This means the graph will be a horizontal line through y = 8.
Drawing this function:
  • Choose various x-values, e.g., x = -2, 0, 4.
  • Compute f(x): \( f(-2) = 8, f(0) = 8, f(4) = 8 \)
Plotting these points, you get (-2, 8), (0, 8), and (4, 8). If you connect these points, the result is a horizontal line at y = 8.
This graphical representation is simple but powerful, showcasing how the function behaves and providing quick insights into its properties, such as constancy and slope.

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Most popular questions from this chapter

Solve each system. If a system's equations are dependent or if there is no solution, state this. $$ \begin{aligned} &x+y+z=83\\\ &y=2 x+3\\\ &z=40+x \end{aligned} $$

Determine whether each pair of lines is parallel, perpendicular, or neither. [ 2.4] $$ \begin{aligned} &3 x-y=12\\\ &y=3 x+7 \end{aligned} $$

Bing Boing Hobbies is willing to produce 100 yo-yo's at \(\$ 2.00\) each and 500 yo-yo's at \(\$ 8.00\) each. Research indicates that the public will buy 500 yo- yo's at \(\$ 1.00\) each and 100 yo-yo's at \(\$ 9.00\) each. Find the equilibrium point.

Solve. $$ \begin{aligned} &\frac{x+2}{3}-\frac{y+4}{2}+\frac{z+1}{6}=0\\\ &\frac{x-4}{3}+\frac{y+1}{4}-\frac{z-2}{2}=-1\\\ &\frac{x+1}{2}+\frac{y}{2}+\frac{z-1}{4}=\frac{3}{4} \end{aligned} $$

Solve each system. If a system's equations are dependent or if there is no solution, state this. $$ \begin{aligned} 3 x+4 y-3 z &=4 \\ 5 x-y+2 z &=3 \\ x+2 y-z &=-2 \end{aligned} $$
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