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Chapter 4: Problem 84

For the following exercises, sketch the graph of each equation. $$ r(x)=4 $$

Short Answer

Expert verified
The graph of \( r(x) = 4 \) is a horizontal line at \( y = 4 \).

Step by step solution

01

Identify the Type of Graph

The equation provided is \( r(x) = 4 \). This is a constant function, where \( r(x) \) is the same for all values of \( x \). A constant function graphs to a horizontal line in the xy-plane.
02

Determine the Line Position

Since the constant value is 4, the horizontal line will be placed parallel to the x-axis at the level where the y-coordinate is 4. This means any point on this graph will have its y-coordinate as 4.
03

Plot the Graph

To sketch the graph, draw a straight horizontal line passing through the point (0,4) on the y-axis. Extend this line to both the left and right as the graph continues indefinitely. This line is parallel to the x-axis.
04

Label the Graph

Label the line with the equation \( r(x) = 4 \) to show that it represents the function given in the exercise.

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Key Concepts

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

Graphing
Graphing a function involves creating a visual representation of the relationship between variables or values. This task uses coordinate points in the Cartesian plane to visually express a given equation. For the function \( r(x) = 4 \), graphing simplifies to plotting a horizontal line where all the y-values are constant.
Understanding graphing is fundamental because it helps in visualizing functions and relationships, making them easier to comprehend at a glance. It visually demonstrates how for each x-coordinate, the y-coordinate is fixed at 4 in a constant function.
  • Choose an equation or function as your starting point.
  • Identify the type of graph: is it linear, quadratic, constant, etc.?
  • Determine key features, such as slope, intercepts, and the general shape.
  • Plot points and draw the graph accordingly.
In this exercise, you'll find that a constant function produces a simple and straightforward graph.
Horizontal Line
A horizontal line on a graph is one where the y-value remains consistent regardless of changing x-values. In the context of the function \( r(x) = 4 \), the horizontal line represents every point where the y-value is 4. This is a defining characteristic of constant functions. By drawing this line parallel to the x-axis, the function clearly shows that x does not impact the line's position vertically.
To graph a horizontal line:
  • Identify the constant y-value. For \( r(x) = 4 \), y is always 4.
  • Draw a straight line across the y-coordinate equal to the constant value, in this case, y = 4.
  • Ensure the line spans the entire graph in both directions horizontally.
This method shows you the stability of the function, as changes in x do not affect the outcome: a constant y-value.
Constant Value
In mathematical functions, a constant value signifies that the output (often y) remains unchanged as the input (often x) varies. In \( r(x) = 4 \), 4 is the constant value. Since the equation lacks an x-term, the output is not influenced by x, manifesting as a horizontal line on a graph.
Constant values make graphing particularly straightforward:
  • Identify the constant from the equation.
  • It represents the unchanging y-value for any x-coordinate.
  • The visual outcome is a horizontal line along the identified constant y-value, spanning the x-axis.
This constant signifies reliability and predictability, with every point along the line yielding the same result.
xy-plane
The xy-plane is the basis for plotting equations and graphing functions, defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin (0,0), dividing the plane into four quadrants.
For the equation \( r(x) = 4 \), utilizing the xy-plane helps effectively visualize the function as a horizontal line. The line runs parallel to the x-axis, intersecting the y-axis at the constant y-value, which is 4.
Key aspects of the xy-plane include:
  • Understanding that the x-axis runs horizontally and the y-axis runs vertically.
  • Recognizing that each point on the plane is represented as a pair \((x, y)\).
  • Acknowledging that the plane offers a comprehensive way to visualize equations and their relationships.
Using the xy-plane, students can visualize how equations function graphically, supporting better comprehension of mathematical principles.

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Most popular questions from this chapter

For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data. $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 5 & 10 & 15 \\ \hline \boldsymbol{g}(\boldsymbol{x}) & 5 & -10 & -25 & -40 \\ \hline \end{array} $$

In 2003 , a town's population was \(1,431 .\) By 2007 the population had grown to \(2,134\) . Assume the population is changing linearly. a. How much did the population grow between the year 2003 and 2007\(?\) b. How long did it take the population to grow from \(1,431\) people to \(2,134\) people? c. What is the average population growth per year? d. What was the population in the year 2000 ? e. Find an equation for the population, \(P\) of the town \(t\) years after 2000 . f. Using your equation, predict the population of the town in 2014 .

A clothing business fi ds there is a linear relationship between the number of shirts, \(n,\) it can sell and the price, \(p\), it can charge per shirt. In particular, historical data shows that 1,000 shirts can be sold at a price of \(\$ 30,\) while 3,000 shirts can be sold at a price of \(\$ 22 .\) Find a linear equation in the form \(p(n)=m n+b\) that gives the price \(p\) they can charge for \(n\) shirts.

For the following exercises, find the \(x\) - and \(y\) -intercepts of each equation. $$ g(x)=2 x+4 $$

In \(2004,\) a school population was \(1,001\) . By 2008 the population had grown to \(1,697\) . Assume the population is changing linearly. a. How much did the population grow between the year 2004 and 2008\(?\) b. How long did it take the population to grow from \(1,001\) students to \(1,697\) students? c. What is the average population growth per year? d. What was the population in the year 2000\(?\) e. Find an equation for the population, \(P,\) of the school \(t\) years after 2000 . f. Using your equation, predict the population of the school in 2011.
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