Question

Determine whether the graphs of the following equations and functions have symmetry about the \(x\) -axis, the \(y\) -axis, or the origin. Check your work by graphing. $$f(x)=x^{4}+5 x^{2}-12$$

Short answer

Answer: The function has symmetry about the y-axis.

Step-by-step solution

Determine symmetry about the x-axis

To check for symmetry about the x-axis, we need to verify if \(f(x)=f(-x)\). Let's substitute \(-x\) in the function and see if it is equivalent to \(f(x)\). $$ f(-x) = (-x)^4 + 5(-x)^2 - 12 \\ =(-x)^4 -5x^2 -12 $$ Comparing the given function with this expression, we can see that: $$ f(x) = x^4 + 5x^2 - 12 \neq (-x)^4 -5x^2 -12 = f(-x) $$ Thus, the function does not have symmetry about the x-axis.

Determine symmetry about the y-axis

To check for symmetry about the y-axis, we need to verify if \(f(-x)=f(x)\). Let's substitute \(-x\) in the function and see if it is equivalent to \(f(x)\). $$ f(-x) = (-x)^4 + 5(-x)^2 - 12 \\ =(-x)^4 + 5x^2 -12 $$ Comparing the given function with this expression, we can see that: $$ f(x) = x^4 + 5x^2 - 12 = (-x)^4 + 5x^2 - 12 = f(-x) $$ Thus, the function has symmetry about the y-axis.

Determine symmetry about the origin

To check for symmetry about the origin, we need to verify if \(f(-x)=-f(x)\). Let's substitute \(-x\) in the function and verify if it is equivalent to \(-f(x)\). We already have the expression for \(f(-x)\) from step 2: $$ f(-x) = (-x)^4 + 5x^2 - 12 $$ Now we need to find \(-f(x)\): $$ -f(x) = -(x^4 + 5x^2 - 12) \\ =-x^4 - 5x^2 + 12 $$ Comparing the expressions for \(f(-x)\) and \(-f(x)\), we can see that: $$ f(-x) = (-x)^4 + 5x^2 - 12 \neq -x^4 - 5x^2 + 12 = -f(x) $$ Thus, the function does not have symmetry about the origin.

Conclusion

After analyzing the given function, we can conclude that it only has symmetry about the y-axis and does not have symmetry about the x-axis or the origin.

Key concepts

Even and Odd Functions

Understanding even and odd functions is key to grasping symmetry in graphs. Functions are classified as even, odd, or neither based on their symmetry properties. An even function is one where replacing the variable with its negative (\(-x\)) does not change the function: \(f(x) = f(-x)\). This characteristic implies symmetry about the y-axis.

An odd function, on the other hand, boasts symmetry about the origin. For these functions, replacing the variable with its negative should result in the function being the negative of the original function: \(f(-x) = -f(x)\). This means if you rotate the graph 180 degrees around the origin, it will look the same.

To find whether a function is even, odd, or neither, substitute \(-x\) in place of \(x\) in the function and compare the results. If neither condition holds, then the function doesn't have these classical symmetries.

Graphical Symmetry

Graphical symmetry in functions refers to how the graph of a function behaves when reflected or rotated.
It’s a handy concept for predicting the shape and behavior of graphs without plotting every point.

• Symmetry about the y-axis: If the function \(f(x)\) is equal to \(f(-x)\), then the graph is mirrored along the y-axis.
• Symmetry about the x-axis: For symmetry about the x-axis, each point (y, x) on the graph must have a corresponding point (-y, x) on the graph. However, this symmetry is less common in basic functions because it generally doesn’t pass the vertical line test (and thus not considered a function).
• Symmetry about the Origin: This occurs if \(f(-x) = -f(x)\), implying that what's happening at one point is mirrored diagonally through the origin.

Understanding these symmetries can simplify graphing and provide insights into the function's nature.

Checking Symmetry of Functions

To check if a function exhibits symmetry, you can follow some systematic steps. First, decide the type of symmetry you are checking for: y-axis, x-axis, or the origin. For each type, perform the necessary substitutions and compare results:

• Substitution for y-axis symmetry: Calculate \(f(-x)\) and compare it with \(f(x)\). If they're equal, symmetry about the y-axis is verified.
• Substitution for x-axis symmetry: Check if the equation holds true for both \(f(x)\) and \(-f(x)\). This isn't common for functions because it requires every input to have two possible outputs.
• Substitution for origin symmetry: Calculate \(f(-x)\) and \(-f(x)\). If they're equal, the graph is symmetric about the origin.

By employing these tests strategically, you can effectively reveal the symmetry properties of the given function.