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Chapter 3: Problem 50

Complete each table of function values and then graph each function. See Examples 5 and \(6 .\) $$ h(x)=|x+2| $$ $$ \begin{array}{|r|l|} \hline \boldsymbol{x} & \boldsymbol{h}(\boldsymbol{x}) \\ \hline-5 & \\ -4 & \\ -3 & \\ -2 & \\ -1 & \\ 0 & \\ 1 & \\ \hline \end{array} $$

Short Answer

Expert verified
The completed table for \( h(x) = |x+2| \) is: (-5,3), (-4,2), (-3,1), (-2,0), (-1,1), (0,2), (1,3). Graph forms a V-shape.

Step by step solution

01

Find h(x) for x = -5

Substitute \( x = -5 \) into the function \( h(x) = |x + 2| \). We get \( h(-5) = |-5 + 2| = |-3| = 3 \).
02

Find h(x) for x = -4

Substitute \( x = -4 \) into the function. \( h(-4) = |-4 + 2| = |-2| = 2 \).
03

Find h(x) for x = -3

Substitute \( x = -3 \) into the function. \( h(-3) = |-3 + 2| = |-1| = 1 \).
04

Find h(x) for x = -2

Substitute \( x = -2 \) into the function. \( h(-2) = |-2 + 2| = |0| = 0 \).
05

Find h(x) for x = -1

Substitute \( x = -1 \) into the function. \( h(-1) = |-1 + 2| = |1| = 1 \).
06

Find h(x) for x = 0

Substitute \( x = 0 \) into the function. \( h(0) = |0 + 2| = |2| = 2 \).
07

Find h(x) for x = 1

Substitute \( x = 1 \) into the function. \( h(1) = |1 + 2| = |3| = 3 \).
08

Complete the Table

Now fill in the table with the values calculated:\[\begin{array}{|r|l|}\hline \boldsymbol{x} & \boldsymbol{h}(oldsymbol{x}) \\hline-5 & 3 \-4 & 2 \-3 & 1 \-2 & 0 \-1 & 1 \0 & 2 \1 & 3 \\hline\end{array}\]
09

Graph h(x)

Plot the points from the table on a coordinate grid: (-5,3), (-4,2), (-3,1), (-2,0), (-1,1), (0,2), (1,3). Connect the points with straight lines to reflect the V-shape of the absolute value function.

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Key Concepts

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

Function Table
A function table is a simple way to organize values of a function. It lists input values, which are the values of \( x \), and their corresponding output values, generated by the function formula. For the absolute value function \( h(x) = |x + 2| \), we substitute each \( x \)-value to find the \( h(x) \) output.

To fill the given function table, follow these steps:
  • Substitute each given \( x \)-value into the function \( h(x) = |x + 2| \).
  • Calculate the absolute value to find \( h(x) \) for each \( x \).
For instance:
  • For \( x = -5 \), \( h(-5) = |-5 + 2| = 3 \)
  • Continue this process for all \( x \) values in the table.
This method ensures all function outputs are clearly paired with their inputs in an organized manner.
Graphing
Graphing an absolute value function involves plotting the function's output values on the coordinate plane. From the function table, you have pairs of corresponding \( x \) and \( h(x) \) values.

Here's how to graph:
  • Plot each \( (x, h(x)) \) point from the table.
  • For instance, the point \((-5, 3)\) corresponds to \( x = -5 \) and \( h(x) = 3 \).
  • Repeat for all points such as \((-4, 2)\), \((-3, 1)\), etc.
After plotting all the points, connect them with lines. For an absolute value function, these lines will form a 'V' shape, reflecting symmetry around the lowest point, or the vertex of the graph. This V-shaped representation is characteristic of absolute value functions.
Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is essential for graphing functions. It consists of two perpendicular axes. The horizontal axis is called the \( x\)-axis and the vertical axis is the \( y\)-axis. These axes intersect at a point known as the origin, which has the coordinate \( (0,0) \).

When plotting the absolute value function \( h(x) = |x + 2| \):
  • Each point \((x, h(x))\) is positioned on the coordinate plane based on its \( x \) and \( y \) values.
  • The \( x \)-values are plotted along the horizontal axis.
  • The \( h(x) \)-values are plotted along the vertical axis.
The coordinate plane allows for a visual representation, making it easier to see how values change across different inputs and outputs.
Piecewise Function
An absolute value function can be understood as a type of piecewise function. A piecewise function is defined by multiple sub-functions, with each one applying to different parts of the domain.

For \( h(x) = |x + 2| \), we can view it as:
  • \( h(x) = x + 2 \), when \( x + 2 \geq 0 \).
  • \( h(x) = -(x + 2) \), when \( x + 2 < 0 \).
This means, right up until \( x = -2 \), the function reflects the negative side of the line \( x + 2 \) and then turns into the positive side beyond that point. The graph visually demonstrates this switch, forming a vertex where the change happens. Understanding this piecewise nature helps in analyzing the behavior and properties of absolute value graphs.

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Most popular questions from this chapter

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