Does the equation f(x) = 3x^2 - 4 exhibit symmetry with respect to the x-axis, the y-axis, both, or neither?

Does the equation f(x) = 3x^2 - 4 exhibit symmetry with respect to the x-axis, the y-axis, both, or neither?

• A. X-axis
• B. Y-axis
• C. Both
• D. Neither

Answer: (A)

To determine if the function f(x) = 3x^2 - 4 exhibits symmetry, we need to analyze its behavior in relation to the x-axis and y-axis. For symmetry with respect to the y-axis, a function must satisfy the condition that f(-x) = f(x) for all x in the domain of the function. If we calculate f(-x): f(-x) = 3(-x)^2 - 4 = 3x^2 - 4 = f(x). Since f(-x) equals f(x), the function is symmetric with respect to the y-axis. Now, to check for symmetry with respect to the x-axis, the function would have to satisfy the condition that for every (x, y) point on the curve, the point (x, -y) is also on the curve. This requires solving -f(x) = f(x), which is not the case for our function since y-values will not equal their negatives. Thus, f(x) = 3x^2 - 4 is symmetric with respect to the y-axis, not the x-axis. The correct interpretation of the symmetry of this function indicates that the proper conclusion aligns with the reflected properties of the graph

To determine if the function f(x) = 3x^2 - 4 exhibits symmetry, we need to analyze its behavior in relation to the x-axis and y-axis.

For symmetry with respect to the y-axis, a function must satisfy the condition that f(-x) = f(x) for all x in the domain of the function. If we calculate f(-x):

f(-x) = 3(-x)^2 - 4 = 3x^2 - 4 = f(x).

Since f(-x) equals f(x), the function is symmetric with respect to the y-axis.

Now, to check for symmetry with respect to the x-axis, the function would have to satisfy the condition that for every (x, y) point on the curve, the point (x, -y) is also on the curve. This requires solving -f(x) = f(x), which is not the case for our function since y-values will not equal their negatives.

Thus, f(x) = 3x^2 - 4 is symmetric with respect to the y-axis, not the x-axis. The correct interpretation of the symmetry of this function indicates that the proper conclusion aligns with the reflected properties of the graph