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Magazine Chapter 1: Problem 25
Set up a table to sketch the graph of each function using the following values: \(x=-3,-2,-1,0,1,2,3\) \(\quad f(x)=2|x|\)
Short Answer
Expert verified
The graph is a V-shape with points \((-3, 6), (-2, 4), (-1, 2), (0, 0), (1, 2), (2, 4), (3, 6)\).
Step by step solution
01
Understand the Function
The function given is \( f(x) = 2|x| \). This means for each value of \( x \), you take the absolute value of \( x \), then multiply by 2.
02
Calculate the Values for Negative x
Start with negative values of \( x \):\ - When \( x = -3 \), \( f(-3) = 2| -3 | = 2 \times 3 = 6 \).\ - When \( x = -2 \), \( f(-2) = 2| -2 | = 2 \times 2 = 4 \).\ - When \( x = -1 \), \( f(-1) = 2| -1 | = 2 \times 1 = 2 \).
03
Calculate the Values for x = 0
For \( x = 0 \), the function value is straightforward: \ - \( f(0) = 2 |0| = 0 \).
04
Calculate the Values for Positive x
Proceed with positive values of \( x \):\ - When \( x = 1 \), \( f(1) = 2|1| = 2 \times 1 = 2 \).\ - When \( x = 2 \), \( f(2) = 2|2| = 2 \times 2 = 4 \).\ - When \( x = 3 \), \( f(3) = 2|3| = 2 \times 3 = 6 \).
05
Set Up the Table
Create a table with two columns, one for \( x \) and one for \( f(x) \):\\[\begin{array}{|c|c|}\hline x & f(x) \hline -3 & 6 \ -2 & 4 \ -1 & 2 \ 0 & 0 \ 1 & 2 \ 2 & 4 \ 3 & 6 \hline\end{array}\]
06
Sketch the Graph
Using the table, plot the points \((-3, 6), (-2, 4), (-1, 2), (0, 0), (1, 2), (2, 4), (3, 6)\) on a graph. Notice the V-shape formed due to the absolute value function.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Absolute Value Function
The absolute value function is a special kind of function that outputs the distance of a number from zero on the number line, without considering direction. This means that for any number, whether positive, zero, or negative, the absolute value is always non-negative. The notation is \(|x|\). For example:
- \(|3| = 3\)
- \(|-3| = 3\)
- \(|0| = 0\)
Function Graphing
Graphing functions allows us to visually understand the relationship between variables. To graph the function \( f(x) = 2|x| \), we plot points on a coordinate plane using input-output pairs, where \(x\) is the input and \(f(x)\) is the output. This function is a piecewise function, meaning it operates in one way for \(x < 0\) and another way for \(x \geq 0\).When graphing piecewise functions:
- Calculate several function values to see how the function behaves.
- Plot these points accurately on a graph.
- Connect the points smoothly understanding the general shape expected from the function.
Table of Values
A table of values is an essential tool for graphing functions. It provides a clear and organized way to calculate and list function outputs for selected inputs. For the function \( f(x) = 2|x| \), generating a table helps us track how outputs change with inputs like \(-3, -2, -1, 0, 1, 2,\) and \(3\).Here's a breakdown of the process:
- Choose a range of \(x\) values suitable for your graph.
- Calculate the corresponding \(f(x)\) values using the function rule, ensuring all steps are clear.
- Fill the table, pairing each \(x\) with its \(f(x)\) result.
V-shaped Graph
A V-shaped graph is a hallmark of an absolute value function, characterized by its symmetry and sharp turning point at the vertex. For the function \( f(x) = 2|x| \), the V-shaped graph reflects how the absolute value component affects the geometry of the graph.Key features of a V-shaped graph include:
- The vertex, which is the lowest point, located at \((0, 0)\) for our function.
- Two rays emanating from the vertex, one for \(x < 0\) and one for \(x > 0\), merging at the vertex.
- A symmetric shape about the y-axis, ensuring that points such as \((-3,6)\) and \((3,6)\) lie on opposing sides of the y-axis at equal distances.
