More Algebra 2 problems.

__August 2017, Part II__

All Questions in Part II are worth 2 credits. Work need be shown (or explained or justified) for full credit. Correct numerical answers with no work receive one credit.

*31. Algebraically determine whether the function j(x)= x ^{4} - 3x^{2} - 4 is odd, even, or neither.
*

**Answer: **

A function is *even* if it is symmetrical about the y-axis. If reflected across the y-axis, it will be mapped onto itself. Because of this symmetry, for any value of x, f(x) = f(-x).

A function is *odd* if it is rotational symmetrical about the origin. If rotated 180 degrees about the origin, it will be mapped onto itself. Because of this symmetry, for any value of x, f(-x) = -f(x).

All the exponents (including the constant) have even exponents, so we know that the function is even. We need to show it *algebraically*, but knowing this tells you which situation you should start with.

f(-x) = (-x)^{4} - 3(-x)^{2} - 4

f(-x) = x^{4} - 3x^{2} - 4

f(-x) = f(x), therefore f(x) is even.

*32. On the axes below, sketch a possible function p(x) = (x - a)(x - b)(x + c), where a, b, and c are
positive, a > b, and p(x) has a positive y-intercept of d. Label all intercepts.
*

**Answer: **

In the given function, a and b are roots on the right side of the y-axis (positive), and -c will be on left side (negative). Also, b comes before a because a > b. Finally, d is on the y-axis somewhere above the x-axis. This is just a sketch, it doesn't have to be perfect. At a minimum, please make sure your sketch passes the vertical line test -- don't be sloppy.

The line has to go from -c to d, so it starts in Quadrant III, through -c to d, then down to b and back up through a and beyond.

See the image below.

Note: *d* does not have to be a local maximum, but it could be. And *c* is a positive number, so the axis must be labeled *-c*.

Comments and questions welcome.

More Algebra 2 problems.