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Chapter 2: Problem 117

From an equation in \(x\) and \(y\), explain how to determine whether the graph of the equation is symmetric with respect to the \(x\) -axis, \(y\) -axis, or origin.

Short Answer

Expert verified
Test for symmetry by replacing variables: x-axis (\text{replace } y \text{ with } -y), y-axis (\text{replace } x \text{ with } -x), and origin (\text{replace } x \text{ and } y \text{ with } -x \text{ and } -y).

Step by step solution

01

Check Symmetry with Respect to the x-axis

To determine if the graph of the equation is symmetric with respect to the x-axis, replace every instance of y with -y in the equation. If the resulting equation is equivalent to the original equation, then the graph is symmetric with respect to the x-axis.
02

Check Symmetry with Respect to the y-axis

To check symmetry with respect to the y-axis, replace every instance of x with -x in the equation. If the resulting equation is the same as the original equation, then the graph is symmetric with respect to the y-axis.
03

Check Symmetry with Respect to the Origin

For symmetry with respect to the origin, replace both x with -x and y with -y in the equation. If the resulting equation matches the original equation, then the graph is 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.

x-axis symmetry
To determine if a graph is symmetric with respect to the x-axis, you need to perform a simple test.
Replace every instance of y with -y in the equation. Here's an example to make it clear:
  • Original Equation: \(y = x^2 + 3\)
  • Replace y with -y: \(-y = x^2 + 3\)
If the new equation, after replacing y with -y, is the same as the original equation, the graph is symmetric with respect to the x-axis.

For our example, \(-y = x^2 + 3\) is not the same as \(y = x^2 + 3\).

This means the graph is not symmetric with respect to the x-axis.

This test can save you lots of time and prevent potential missteps when plotting graphs.
y-axis symmetry
Checking for y-axis symmetry is quite similar.
Here, you replace every instance of x with -x:
  • Original Equation: \(y = x^2 + 3\)
  • Replace x with -x: \(y = (-x)^2 + 3\)
If the resulting equation is the same as the original equation, the graph will be symmetric with respect to the y-axis.

In our example, \(y = (-x)^2 + 3 = x^2 + 3\) matches the original equation.

Hence, the graph of \(y = x^2 + 3\) is symmetric with respect to the y-axis.

This symmetry is especially useful when dealing with even functions where \(f(x) = f(-x)\).
origin symmetry
Origin symmetry requires a bit more work, but don't worry, it's straightforward.
Here, you replace both x with -x and y with -y:
  • Original Equation: \(y = x^2 + 3\)
  • Replace x with -x and y with -y: \(-y = (-x)^2 + 3\)
If the new equation is still the same as the original, the graph has origin symmetry.

Checking for our example:
\(-y = x^2 + 3\) is NOT the same as \(y = x^2 + 3\).

Therefore, the graph does not have origin symmetry.

This quality is characteristic of odd functions where \(f(-x) = -f(x)\). By testing these simple replacements, you can easily determine the symmetry of any given graph.

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Most popular questions from this chapter

A function is given. (See Examples \(4-5)\) a. Find \(f(x+h)\). b. Find \(\frac{f(x+h)-f(x)}{h}\). $$f(x)=x^{2}-3 x$$

Refer to the functions \(f, g,\) and \(h\) and evaluate the given functions. \(f(x)=2 x+1 \quad g(x)=x^{2} \quad h(x)=\sqrt[3]{x}\) $$(h \circ g \circ f)(x)$$

Use a graphing utility to graph the piecewise-defined function. $$ \begin{aligned} &z(x)=\left\\{\begin{array}{ll} 2.5 x+8 & \text { for } x<-2 \\ -2 x^{2}+x+4 & \text { for }-2 \leq x<2 \\ -2 & \text { for } x \geq 2 \end{array}\right.\\\ &\text { Is there actually a "gap" in the graph at } x=2 ? \end{aligned} $$

Graph the function. $$ k(x)=\left\\{\begin{aligned} 3 x & \text { for } x<1 \\ -3 x & \text { for } x \geq 1 \end{aligned}\right. $$

Use a graphing utility to approximate the relative maxima and relative minima of the function on the standard viewing window. Round to 3 decimal places. $$ g(x)=0.4 x^{2}-3 x-2.2 $$
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