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Chapter 2: Problem 16

Determine whether the graph of the equation is symmetric with respect to the \(x\) -axis, \(y\) -axis, origin, or none of these. $$ x=y^{2}+3 $$

Short Answer

Expert verified
The graph is symmetric with respect to the x-axis.

Step by step solution

01

Rewriting the Original Equation

The given equation is \(x = y^2 + 3\). Write it down so that it's clear for transformations.
02

Test for Symmetry with Respect to the x-axis

Replace every \(y\) with \(-y\) in the equation and see if we get the same equation. We get \(x = (-y)^2 + 3 = y^2 + 3\), which is the same as the original equation. Therefore, the graph is symmetric with respect to the x-axis.
03

Test for Symmetry with Respect to the y-axis

Replace every \(x\) with \(-x\) and see if we get the same equation. We get \(-x = y^2 + 3\), which is not the same as the original equation. Therefore, the graph is not symmetric with respect to the y-axis.
04

Test for Symmetry with Respect to the Origin

Replace \(x\) with \(-x\) and \(y\) with \(-y\). We get \(-x = (-y)^2 + 3 = y^2 + 3\), which is not the same as the original equation. Therefore, the graph is not symmetric with respect to the origin.

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Key Concepts

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

symmetric with respect to x-axis
Understanding symmetry with respect to the x-axis can greatly simplify working with and visualizing graphs. Imagine flipping the graph over the x-axis like a mirror. If the graph looks the same after this flip, then it is symmetric with respect to the x-axis.

How to test:
  • Replace every instance of y with -y in the equation. This means performing a substitution: if the original equation is f(x, y), then you substitute y with -y to get f(x, -y).
  • Check if the new equation matches the original equation. If they are the same, then the graph is symmetric with respect to the x-axis.
In the exercise, the equation given is x = y^2 + 3. Replacing y with -y, we get x = (-y)^2 + 3. Simplifying this, we get back x = y^2 + 3, which matches the original equation. Hence, the graph is symmetric with respect to the x-axis.
symmetric with respect to y-axis
When a graph is symmetric with respect to the y-axis, it means the graph remains unchanged if reflected over the y-axis. Visualize folding the graph along the y-axis; if both halves align perfectly, then it is symmetric with respect to the y-axis.

How to test:
  • Replace every instance of x in the equation with -x. If the original equation is f(x, y), you change it to f(-x, y).
  • Compare the new equation to the original. If they match, the graph is symmetric with respect to the y-axis.
Looking at our example equation x = y^2 + 3, we replace x with -x. This gives us -x = y^2 + 3, which does not match the original equation. Thus, the graph is not symmetric with respect to the y-axis.
symmetric with respect to origin
A graph that is symmetric with respect to the origin looks the same if rotated 180 degrees around the origin. This symmetry involves both axes.

How to test:
  • Replace both x with -x and y with -y. For an equation f(x, y), it transforms to f(-x, -y).
  • If this new equation matches the original, the graph is symmetric with respect to the origin.
In our equation x = y^2 + 3, substitute x with -x and y with -y. This gives us -x = (-y)^2 + 3, which simplifies to -x = y^2 + 3. Since the newly formed equation is different from the original, the graph is not symmetric with respect to the origin.

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