Algebra 2 Problems of the Day (open ended)

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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)= x⁴ - 3x² - 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) = x⁴ - 3x² - 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.

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