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Chapter 4: Problem 29

Graph each function by making a table of values and plotting points. $$h(x)=-3$$

Short Answer

Expert verified
Create a table of values for the given function \(h(x) = -3\), which yields the same output -3 for different \(x\) values: | x | h(x) | |----|------| |-2 | -3 | |-1 | -3 | | 0 | -3 | | 1 | -3 | | 2 | -3 | Plot these points on a coordinate plane and connect them to form a horizontal line that intersects the y-axis at -3. The graph of the function \(h(x)=-3\) is a horizontal line.

Step by step solution

01

Create the table of values

Choose several different values of \(x\) and calculate the corresponding values of the function. In the case of \(h(x)=-3\), the function value will always be -3, no matter the value of \(x\). Here is a table of values for the function: | x | h(x) | |----|------| |-2 | -3 | |-1 | -3 | | 0 | -3 | | 1 | -3 | | 2 | -3 | As we can see, the value of \(h(x)\) remains constant at -3 for all values of \(x\).
02

Plotting Points

Now let's plot the points from the table on a coordinate plane: 1. Plot the point `A` at (-2, -3) 2. Plot the point `B` at (-1, -3) 3. Plot the point `C` at (0, -3) 4. Plot the point `D` at (1, -3) 5. Plot the point `E` at (2, -3)
03

Draw the Graph

Since all of the points for the function lie on the same horizontal line, connect the points using a straight line. The line should extend infinitely in both directions as the function is defined for all values of \(x\). The graph of the function \(h(x) = -3\) is a horizontal line that intersects the y-axis at -3.

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Key Concepts

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

Constant Function
A constant function is a type of linear function where the output value remains the same regardless of the input value. It's simple to identify because the function's formula takes the form \( f(x) = c \), where \( c \) is a fixed number.
  • No matter what \( x \) value you plug into the equation, the output remains constant. For instance, in the function \( h(x) = -3 \), the result will always be -3, whether \( x \) is 1, 10, or 1,000.
  • This behavior makes constant functions straightforward because it requires no calculations after obtaining \( c \).
Think of a constant function as a calm beach on a sunny day鈥攊t doesn't change, providing the same experience every time.
Horizontal Line
When you graph a constant function, you'll see a horizontal line on the coordinate plane. This line represents the unchanged output of the function.
  • A horizontal line extends sideways without any vertical changes.
  • In the graph of \( h(x) = -3 \), it cuts through all points where \( y \) equals -3.
Imagine it as a flat road stretching infinitely, showing that wherever you go horizontally, you stay at the same altitude vertically.
Table of Values
Creating a table of values is a useful step when graphing. It helps organize your inputs and their corresponding outputs.
  • For a constant function like \( h(x) = -3 \), you choose different \( x \) values, and the \( y \) value remains -3 for each.
  • It can guide you in plotting points accurately on a graph.
Picture a table of values as a grocery list. It allows you to track and ensure you have all the necessary elements to plot the graph correctly.
Coordinate Plane
The coordinate plane is a two-dimensional space where functions are graphed. It consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).
  • X-axis: Represents the input values, often displayed horizontally.
  • Y-axis: Represents the output values, shown vertically.
  • The point where these axes intersect is known as the origin, labeled (0, 0).
On the coordinate plane, each point is identified by a pair of numbers, called coordinates, such as (x, y). It allows you to visualize mathematical functions and see the relationships between variables in a clear and structured way.

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Most popular questions from this chapter

since 1998 the population of Maine has been increasing by about 8700 people per year. In 2001 , the population of Maine was about \(1,284,000 .\) a) Write a linear equation to model this data. Let \(x\) represent the number of years after \(1998,\) and let \(y\) represent the population of Maine. b) Explain the meaning of the slope in the context of the problem. c) According to the equation, how many people lived in Maine in \(1998 ?\) in \(2002 ?\) d) If the current trend continues, in what year would the population be \(1,431,900 ?\)

A \(16 \times\) DVD recorder can transfer \(21.13 \mathrm{MB}\) (megabytes) of data per second onto a recordable DVD. The function \(D(t)=21.13 t\) describes how much data, \(D\) (in megabytes), is recorded on a DVD in \(t\) sec. (Source: www.osta.org) a) How much data is recorded after 12 sec? b) How much data is recorded after 1 min? c) How long would it take to record \(422.6 \mathrm{MB}\) of data? d) Graph the function.

Lines \(L_{1}\) and \(L_{2}\) contain the given points. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. $$\begin{aligned}&L_{1}:(-5,4),(-5,-1)\\\&L_{2}:(-1,2),(-3,2)\end{aligned}$$

Write an equation of the line parallel to the given line and containing the given point. Write the answer in slope intercept form or in standard form, as indicated. $$y=\frac{1}{2} x-5 ;(4,5) ; \text { standard form }$$

Write an equation of the line parallel to the given line and containing the given point. Write the answer in slope intercept form or in standard form, as indicated. \(y=4 x+9 ;(0,2) ;\) slope-intercept form
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