Final answer:
The equation 4x² - y + 2z² = 0 is a parabolic cylinder with the y-term isolated, indicative of parabolas that open along the y-axis. The surface can be sketched as an infinitely long cylinder with parabolic cross-sections.
Step-by-step explanation:
To reduce the equation to one of the standard forms, we need to complete the square for the y-term. However, since there is no y2 term and the coefficient of y is -1, we could simply move the y term to the other side to complete the square.
The given equation is 4x2 - y + 2z2 = 0. To start the process of standardization, we can rewrite it as:
y = 4x2 + 2z2
This is a quadratic equation in x and z, which is a parabolic cylinder. The axis of this parabolic cylinder is aligned with the y-axis because the y-term is linear and isolated on one side.
Without specific scale factors, we can sketch the shape by showing a set of parabolas opening along the y-axis at different x-values and, similarly, along the z-axis at different y-values. This surface will have an infinitely long cylindrical shape with cross-sections shaped like parabolas.