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Magazine Chapter 1: Problem 15
Determine all intercepts of the graph of the equation. Then decide whether the graph is symmetric with respect to the \(x\) axis, the \(y\) axis, or the origin. $$ \sqrt{x}+\sqrt{y}=1 $$
Short Answer
Expert verified
Intercepts are (1,0) and (0,1); no symmetry.
Step by step solution
01
- Find the x-intercept
To find the x-intercept, set \(y=0\) in the equation \(\sqrt{x} + \sqrt{y} = 1\). This gives \(\sqrt{x} + \sqrt{0} = 1\), simplifying to \(\sqrt{x} = 1\). Solving for \(x\), we square both sides to obtain \(x = 1\). Thus, the x-intercept is \((1, 0)\).
02
- Find the y-intercept
To find the y-intercept, set \(x=0\) in the equation \(\sqrt{x} + \sqrt{y} = 1\). This gives \(\sqrt{0} + \sqrt{y} = 1\), simplifying to \(\sqrt{y} = 1\). Solving for \(y\), we square both sides to obtain \(y = 1\). Thus, the y-intercept is \((0, 1)\).
03
- Test for symmetry with respect to x-axis
To check for symmetry with respect to the x-axis, replace \(y\) with \(-y\) in the equation to get \(\sqrt{x} + \sqrt{-y} = 1\). Since \(\sqrt{-y}\) is not a real number for \(y \ge 0\), the equation does not satisfy a real function, meaning it is not symmetric about the x-axis.
04
- Test for symmetry with respect to y-axis
To check for symmetry with respect to the y-axis, replace \(x\) with \(-x\) in the equation to get \(\sqrt{-x} + \sqrt{y} = 1\). Since \(\sqrt{-x}\) is not a real number for \(x \ge 0\), the equation does not satisfy a real function, meaning it is not symmetric about the y-axis.
05
- Test for symmetry with respect to the origin
To check for symmetry with respect to the origin, replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation: \(\sqrt{-x} + \sqrt{-y} = 1\). Since both \(\sqrt{-x}\) and \(\sqrt{-y}\) are not real numbers for positive non-zero \(x\) and \(y\), the equation does not satisfy a real function, meaning it is not symmetric about the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
x-intercept
The x-intercept of a graph is the point where the graph crosses the x-axis. At this point, the value of the y-coordinate is zero. To find the x-intercept for the equation \( \sqrt{x} + \sqrt{y} = 1 \), we set \( y = 0 \). This modification simplifies the equation to \( \sqrt{x} = 1 \).
Solving for \( x \), we square both sides, resulting in \( x = 1 \). Thus, the x-intercept of the given graph is at the point \((1, 0)\). This means that when you look at the graph, it touches the x-axis at this coordinate.
Knowing the x-intercept helps in sketching the graph and understanding the behavior of the function near the horizontal axis.
Solving for \( x \), we square both sides, resulting in \( x = 1 \). Thus, the x-intercept of the given graph is at the point \((1, 0)\). This means that when you look at the graph, it touches the x-axis at this coordinate.
Knowing the x-intercept helps in sketching the graph and understanding the behavior of the function near the horizontal axis.
y-intercept
The y-intercept is the point where the graph crosses the y-axis, meaning the x-coordinate is zero at this point. To find the y-intercept of the equation \( \sqrt{x} + \sqrt{y} = 1 \), we set \( x = 0 \). This gives the equation \( \sqrt{y} = 1 \).
To solve for \( y \), we square both sides to obtain \( y = 1 \). Therefore, the y-intercept is \((0, 1)\). This indicates where the graph intersects the y-axis.
This information is crucial for graphing, as it tells us how the function behaves as it approaches the vertical axis and gives clues about the graph's starting point on the y-axis.
To solve for \( y \), we square both sides to obtain \( y = 1 \). Therefore, the y-intercept is \((0, 1)\). This indicates where the graph intersects the y-axis.
This information is crucial for graphing, as it tells us how the function behaves as it approaches the vertical axis and gives clues about the graph's starting point on the y-axis.
symmetric about x-axis
Graph symmetry about the x-axis means that the graph would look the same if flipped over the x-axis.
To test our equation \( \sqrt{x} + \sqrt{y} = 1 \) for this type of symmetry, we replace \( y \) with \( -y \).
This gives us \( \sqrt{x} + \sqrt{-y} = 1 \).
However, \( \sqrt{-y} \) is not defined in the real number system for \( y \ge 0 \).
Therefore, the equation does not describe a real function, indicating that it is not symmetric about the x-axis.
Understanding this concept helps predict whether a graph repeats across the x-axis or features different behavior.
To test our equation \( \sqrt{x} + \sqrt{y} = 1 \) for this type of symmetry, we replace \( y \) with \( -y \).
This gives us \( \sqrt{x} + \sqrt{-y} = 1 \).
However, \( \sqrt{-y} \) is not defined in the real number system for \( y \ge 0 \).
Therefore, the equation does not describe a real function, indicating that it is not symmetric about the x-axis.
Understanding this concept helps predict whether a graph repeats across the x-axis or features different behavior.
symmetric about y-axis
A graph is symmetric about the y-axis if the left side mirrors the right side. For testing symmetry about the y-axis in the equation \( \sqrt{x} + \sqrt{y} = 1 \), we replace \( x \) with \( -x \).
This results in \( \sqrt{-x} + \sqrt{y} = 1 \).
In this case, \( \sqrt{-x} \) is not defined for \( x \ge 0 \).
This means the equation is not valid in the real number system, and hence the graph is not symmetric about the y-axis.
This analysis tells us whether the graph is likely to mirror along the y-axis, providing insight into the graph's structure.
This results in \( \sqrt{-x} + \sqrt{y} = 1 \).
In this case, \( \sqrt{-x} \) is not defined for \( x \ge 0 \).
This means the equation is not valid in the real number system, and hence the graph is not symmetric about the y-axis.
This analysis tells us whether the graph is likely to mirror along the y-axis, providing insight into the graph's structure.
symmetric about origin
Symmetry about the origin means a 180-degree rotation around the origin keeps the graph unchanged.
Testing for symmetry about the origin involves replacing \( x \) with \(-x\) and \( y \) with \(-y\) in the equation \( \sqrt{x} + \sqrt{y} = 1 \), leading to \( \sqrt{-x} + \sqrt{-y} = 1 \).
Since both \( \sqrt{-x} \) and \( \sqrt{-y} \) are not defined in the real number system, the equation doesn't represent a real function.
Hence, the graph is not symmetric about the origin.
This concept helps visualize whether rotating the graph 180 degrees might result in a repetitive pattern.
Testing for symmetry about the origin involves replacing \( x \) with \(-x\) and \( y \) with \(-y\) in the equation \( \sqrt{x} + \sqrt{y} = 1 \), leading to \( \sqrt{-x} + \sqrt{-y} = 1 \).
Since both \( \sqrt{-x} \) and \( \sqrt{-y} \) are not defined in the real number system, the equation doesn't represent a real function.
Hence, the graph is not symmetric about the origin.
This concept helps visualize whether rotating the graph 180 degrees might result in a repetitive pattern.
