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Magazine Chapter 2: Problem 39
19鈥�44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ y=4-|x| $$
Short Answer
Expert verified
The graph is a downward-opening 'V' with x-intercepts at \((-4, 0)\) and \((4, 0)\), and y-intercept at \((0, 4)\). It's symmetric about the y-axis.
Step by step solution
01
Create a Table of Values
To create a table of values, we need to select different values of \( x \) and compute the corresponding values of \( y \) using the equation \( y = 4 - |x| \). Here's a small table:- If \( x = -5 \), then \( y = 4 - |-5| = 4 - 5 = -1 \).- If \( x = -4 \), then \( y = 4 - |-4| = 4 - 4 = 0 \).- If \( x = -3 \), then \( y = 4 - |-3| = 4 - 3 = 1 \).- If \( x = 0 \), then \( y = 4 - |0| = 4 - 0 = 4 \).- If \( x = 3 \), then \( y = 4 - |3| = 4 - 3 = 1 \).- If \( x = 4 \), then \( y = 4 - |4| = 4 - 4 = 0 \).- If \( x = 5 \), then \( y = 4 - |5| = 4 - 5 = -1 \).This table shows how both positive and negative \( x \) values affect \( y \).
02
Sketch the Graph
Using the table of values, plot each point on a coordinate plane. You should plot points such as \((-5, -1), (-4, 0), (-3, 1), (0, 4), (3, 1), (4, 0), (5, -1)\). Connect these points with straight lines to represent the absolute value function. This graph will resemble a 'V' shape, opening downward with its vertex at the point \((0, 4)\).
03
Find the x-Intercepts
To find the x-intercepts, set \( y = 0 \) in the equation \( y = 4 - |x| \):\[ 0 = 4 - |x| \]\[ |x| = 4 \]This gives \( x = 4 \) or \( x = -4 \). Thus, the x-intercepts are the points \((4, 0)\) and \((-4, 0)\).
04
Find the y-Intercept
To find the y-intercept, set \( x = 0 \) in the equation \( y = 4 - |x| \):\[ y = 4 - |0| = 4 \]Thus, the y-intercept is the point \((0, 4)\).
05
Test for Symmetry
Check for symmetry about the y-axis, x-axis, and origin. The function \( y = 4 - |x| \) is symmetric about the y-axis if \( f(x) = f(-x) \). Since \( |x| = |-x| \), the function is indeed symmetric about the y-axis.For x-axis symmetry: If \( f(x) = -f(x) \), this would need false results in our check. Testing for origin symmetry also fails as \( f(-x) eq -f(x) \), so the graph is only symmetric about the y-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Absolute Value Functions
Absolute value functions are uniquely interesting due to their graph's characteristic V-shape. In the equation \( y = 4 - |x| \), the absolute value expression \(|x|\) affects how the graph behaves.
The absolute value function takes any input \( x \) and makes it non-negative. This translates into a "mirror" effect for negative \( x \) values. When constructing a graph of an absolute value function, you can follow these steps:
The absolute value function takes any input \( x \) and makes it non-negative. This translates into a "mirror" effect for negative \( x \) values. When constructing a graph of an absolute value function, you can follow these steps:
- Create a table of values, using both positive and negative numbers for \( x \).
- Calculate the corresponding \( y \) values by substituting \( x \) values into the equation.
- Plot the points on a coordinate plane and connect them to form the V-shape.
X-intercepts
X-intercepts occur where the graph crosses the x-axis, meaning the \( y \) value is zero. To find these points in the equation \( y = 4 - |x| \), set \( y = 0 \):
\[ 0 = 4 - |x| \]
Solving for \( |x| \) gives 4, leading to \( x = 4 \) or \( x = -4 \).
This tells us the x-intercepts are at the coordinates \((4, 0)\) and \((-4, 0)\).
Intercepts are significant as they provide key reference points on the graph. They show where the graph changes from negative to positive or vice versa, offering insights into the function's behavior across different domains.
\[ 0 = 4 - |x| \]
Solving for \( |x| \) gives 4, leading to \( x = 4 \) or \( x = -4 \).
This tells us the x-intercepts are at the coordinates \((4, 0)\) and \((-4, 0)\).
Intercepts are significant as they provide key reference points on the graph. They show where the graph changes from negative to positive or vice versa, offering insights into the function's behavior across different domains.
Y-intercepts
The y-intercept is where the graph crosses the y-axis, meaning the \( x \) value is zero. In the absolute value function \( y = 4 - |x| \), set \( x = 0 \) to find the y-intercept:
\[ y = 4 - |0| = 4 \]
This results in the y-intercept at the point \((0, 4)\).
Y-intercepts offer a starting point for graphing and help verify the graph's accuracy.
They allow us to understand how the function behaves near the origin. This point, being the highest on the graph, is also the vertex of our V-shaped absolute value function.
\[ y = 4 - |0| = 4 \]
This results in the y-intercept at the point \((0, 4)\).
Y-intercepts offer a starting point for graphing and help verify the graph's accuracy.
They allow us to understand how the function behaves near the origin. This point, being the highest on the graph, is also the vertex of our V-shaped absolute value function.
Symmetry in Graphs
Graphing symmetry thoroughly enhances our understanding of the graph's structure. For the function \( y = 4 - |x| \), symmetry exists particularly about the y-axis. Here's why:
This specific symmetry aids in predicting the function's shape and behavior.
Symmetry plays a crucial role in simplifying graphing and analysis, as it reduces the need to calculate values for all \( x \) - it's only needed for one side of the y-axis.
- Substitute \(-x\) into the function: \( y = 4 - |-x| \).
- Since \(|x| = |-x|\), the function remains unchanged, confirming y-axis symmetry.
This specific symmetry aids in predicting the function's shape and behavior.
Symmetry plays a crucial role in simplifying graphing and analysis, as it reduces the need to calculate values for all \( x \) - it's only needed for one side of the y-axis.
