college algebra word problem

Question

Determine whether there is a minimum or maximum value to the quadratic function.

f (x) = 2x^² − 18x + 8

Find the minimum or maximum value of f.

Find the axis of symmetry.

Answer 1

2X^2 -18x + 8 = 0

divide by 2

x^2 -9x +4 = 0

x^2 - 9x = -4

complete the square, add (9/2)^2 = 81/4 to both sides

x^2 - 9x + 81/4 = -4 + 81/4

factor

(x-9/2)^2 = -4 + 81/4 = -16/4 + 81/4 = 65/4

x-9/2 = sqr(65/4) = (1/2)sqr65

x = 9/2 + or - (1/2) sqr65 = 4.5 + or - 4.03 = 0.47 or 8.53 are the x-intercepts

axis of symmetry is x = 4.5

f(4.5) = 2(4.5)^2 -18(4.5) + 8 = -32.5 is the minimum point (4.5,-32.5)

there is no maximum, as f(x) approaches infinity. It's an upward opening parabola that goes to infinity as x approaches either negative or positive infinity