By the way, to consolidate what you've been given:
1) If y = -4x<sup>2</sup> + kx - 1, determine the value(s) for k which the maximum value of a function is an integer. Explain your reasoning.
You've been told that the maximum of a negative quadratic is the vertex of the corresponding downward-opening parabola. So you need to learn about quadratics, graphing parabolas, and finding vertices.
You've been given the formula for the vertex of a parabola. For f(x) = y = ax<sup>2</sup> + bx + c, the vertex (h, k) is at h = -b/(2a), with k = f(h). So you need to learn how to work with variables, formulas, and function notation.
(Note: The "k" in the vertex formula is not the same "k" as in your exercise. So I'll use "K" in what follows for the vertex-formula "k".)
For this particular exercise, you were given that the maximum value occurs at:
. . . . .h = -k/(2(-4)) = -k/-8 = k/8
This was further explained, step-by-step, to mean that the maximum value is:
. . . . .K = f(h) = -4(k/8)<sup>2</sup> + k(k/8) - 1
. . . . .= -4(k<sup>2</sup>/64) + k<sup>2</sup>/8 - 1
. . . . .= -k<sup>2</sup>/16 + k<sup>2</sup>/8 - 1
. . . . .= k<sup>2</sup>/16 - 1
. . . . .= (k/4)<sup>2</sup> - 1
For this to be a whole-number value, it was explained to you that k has to be divisible evenly by 4, so as to avoid fractions. Since k/4 is squared, of course k could be positive or negative, so the solution, as was provided to you earlier, is "k is any multiple of 4".
(Your tutor can teach you how "k = ±4m for any integer m" means the same thing as the above solution, as you are probably expected to understand this more-technical form).
2. The graph of the function f(x) = x²-kx+k+8 touches the x-axis at one point. What is the value of k?
You were given that the "touching at one point" means that the discriminant of the Quadratic Formula for x<sup>2</sup> - kx + (k + 8) must be equal to zero. So you need to learn how to use the Quadratic Formula, and how to "set things equal to zero".
In this case, you were given the quadratic function:
. . . . .f(x) = x<sup>2</sup> - kx + (k + 8)
You were instructed to find the discriminant of the Quadratic Formula symbolically, and were told to use a = 1, b = -k, and c = k + 8:
. . . . .Let x<sup>2</sup> - kx + (k + 8) = 0
. . . . .discriminant: b<sup>2</sup> - 4ac
. . . . .(-k)<sup>2</sup> - 4(1)(k + 8) = 0
. . . . .k<sup>2</sup> - 4k - 32 = 0
. . . . .(k - 8)(k + 4) = 0
It was then explained that you need to solve the above to find the values of k:
. . . . .k - 8 = 0 or k + 4 = 0
. . . . .k = 8 or k = -4
You were then told that this was the answer.
When you work with your tutor, please print out the various solutions, explanations, and links you'd been given, so you can work together with him to figure these out.
Good luck!
Eliz.