Final answer:
The graph y = 3x^2 - 4 is symmetric with respect to the y-axis because substituting -x into the equation results in the same equation. It is not symmetric with respect to the x-axis or the origin, as substituting -y does not yield the original equation, nor does replacing both x and y with their negatives. Option b.
Step-by-step explanation:
To determine algebraically whether the graph of y = 3x^2 - 4 is symmetric with respect to the x-axis, the y-axis, and the origin, we can apply symmetry tests for each case.
Symmetry with respect to the y-axis: A graph is symmetric about the y-axis if, for every point (x, y) on the graph, the point (-x, y) is also on the graph. For the given equation, if we replace x with -x, we obtain y = 3(-x)^2 - 4, which simplifies to y = 3x^2 - 4, the same as the original equation. Hence, the graph is symmetric about the y-axis.
Symmetry with respect to the x-axis: A graph is symmetric about the x-axis if, for every point (x, y) on the graph, the point (x, -y) is also on the graph. If we replace y with -y, there is no way to manipulate the equation to look like the original, because the negative cannot be factored out from the squared term. Therefore, the graph is not symmetric about the x-axis.
Symmetry with respect to the origin: Origin symmetry means that for every point (x, y) on the graph, the point (-x, -y) is also on the graph. Replacing x with -x and y with -y in the equation gives us -y = 3(-x)^2 - 4, which does not result in the original equation. Consequently, the graph is not symmetric about the origin.
Based on these tests, the correct answer is that the graph of the equation is symmetric with respect to the y-axis but not the x-axis or the origin.
So Option b is the correct answer.