What type of curve(s) does the expression y^2 = 4x^2 - 12 represent?

First, arrange the equation into the general form.

4x² – y² – 12 = 0

From this equation, we observe that A = 4, B = 0, and C = -1.

It is useful to memorize the following properties. If the discriminant B² - 4AC is less than zero, then the conic is a hyperbola. If the discriminant is zero, then conic is a parabola. If the discriminant is greater than zero, the conic is an ellipse.

For this particular case, the conic is a hyperbola.

y² = 4x² + c and y = 2x + c are not the same graph. To see why, square the second expression.

y = 2x + c

y² = (2x + c)²

y² = 4x² + 4xc + c², which is not the same as 4x² + c.

It represents a hyperbola rearrange the terms to get y^2/12 - x^2/3 = 1

This is of the form y^2/a^2 - x^2/b^2 = 1

you will get a parabola symmetric about x-axis that Is two parabolas on the y-axis