The function f is defined as f(x) = 3x2 − kx + 26. What are the two values of k for which f has a minimum value of −1? Enter your answers in the boxes, from least to greatest.

Question

What are the values of k ????

Answer 1

The question asks for value of the "b" coefficient that will give the function which has a minimum value of -1. In other words, the minimum value is the Y coordinate of the vertex of this function, as A = 3 is positive, the parabola points up!

We need to figure out the x coordinate of the vertex. (we have the Y) We can then plug this back into the formula to figure out the value of k, because we know when x is the vertex the Y value is -1.

lets call the x-coordinate of the vertex "v"

Formula for the x coordinate of a vertex is -b/2a (from the quadratic formula) . let this = v.

We know the Y coordinate, (-1) ,so plug this into the original quadratic

Formula 1) -1 = 3v² - k(v) + 26 Formula 2) - (-k) /6 = (v) = k = 6(v)

There are two unknowns ( k and v) and two unique formulas, so we can solve!

The two formulas not in the same form, so let's use substitution.

-1 = 3v² - (6v)v + 26 = -3v² +27 = 0 = 3v² - 27 = 0

Using the same logic as difference of squares, as the v has to cancel out. This factors to (3v+9)(v-3)

Which means two options for x coordinate of the vertex are 3 and -3.

We can then plug those back into the original quadratic to find the value of k.

3(3)² - k(3) + 26 = -1 = 27- 3k -27 3k = 54 k = 18

3(-3)² - k(-3) + 26 = -1 27 + 3k -27 3k = -54 k= -18