Question

A quadratic function has x-intercepts 2 and 6 and its vertex is (4, 8). What is the corresponding quadratic expression?

Answer 1

SOLUTION 1 
If the x-intercepts (roots) are 2 and 6, then it must look like 
y = a(x - 2)(x - 6) 
for some real number a. So you need to find the value of a. They give you the point (x, y) = (4, 8), so plug those numbers in and solve for a. 
y = a(x - 2)(x - 6) with (x, y) = (4, 8) 
8 = a(4 - 2)(4 - 6) 8 = a(2)(-2) 8 = -4a a = -2 
So y = -2(x - 2)(x - 6) y = -2(x² - 8x + 12) y = -2x² + 16x- 24 
SOLUTION 2 
If the vertex is at (4, 8), then it must be of the form 
y = a(x - 4)² + 8 
Since 6 an x-intercept, then (x, y) = (6, 0) is a solution. So we know 
y = a(x - 4)² + 8 with (x, y) = (6, 0) 0 = a(6 - 4)² + 8 0 = 4a + 8 4a = -8 a = -2 
y = -2(x - 4)² + 8 y = -2(x² - 8x + 16) + 8 y = -2x² + 16x -32 + 8 y = -2x² + 16x - 24 
SOLUTION 3 
y = ax² + bx + c We know (x, y) = (2, 0), (6, 0) and (4, 8) are solutions so 
0 = 4a + 2b + c 0 = 36a + 6b + c 8 = 16a + 4b + c 
36a + 6b + c = 0 4a + 2b + c = 0 -------------------------- 32a + 4b = 0 4b = -32a b = -8a 
4a + 2b + c = 0 with b = -8a 4a + 2(-8a) + c = 0 4a -16a + c = 0 -12a + c = 0 c = 12a 
8 = 16a + 4b + c with b = -8a and c = 12a 8 = 16a + 4(-8a) + 12a 8 = 16a - 32a + 12a 8 = -4a a = -2 b = -8a = 16 c = 12a = -24 
y = -2x² + 16x - 24 
SOLUTION 4 They tell you that 2 is an x-intercept. So the polynomial must evaluate to 0 when you let x = 2 
A) 2x^2-16x+24 = 8 - 32 + 24 = 0 possible right answer B) -2x^2+16x-24 = -8 + 32 - 24 = 0 possible right answer C) -2x^2-16x+24 = -8 - 32 + 24 = -16 wrong answer D) -x^2-16x+12 = -4 - 32 + 12 = -24 wrong anser E) x^2-16x-24 = 4 - 32 - 24 = -52 wrong answer 
So A or B is the correct answer. You notice that answer B is the opposite of answer A So x = 6 will probably make both of them evaluate to 0. But they also tell you that (4, 8) is a solution. So, if we let x = 4, the polynomial should evaluate to 8. 
A) 2x^2-16x+24 = 32 - 64 + 24 = -8 wrong answer B) -2x^2+16x-24 
So B must be the right answer. 
SOLUTION 5 
The vertex of y = ax² + bx + c occurs at x = -b/2a They tell you the vertex is at (x, y) = (4, 8) So -b/2a must evaluate to 4. 
A) 2x^2-16x+24 *** -b/2a = 16/4 = 4 possible right answer B) -2x^2+16x-24 *** -b/2a = -16/(-4) = 4 possible right answer C) -2x^2-16x+24 *** -b/2a = 32/(-4) = -8 wrong answer D) -x^2-16x+12 *** -b/2a = 16/(-2) = -8 wrong answer E) x^2-16x-24 *** -b/2a = 32/(-2) = -16 wrong answer 
So A or B is the right answer. Since (4, 8) a vertex, the polynomial should evaluate to 8 if we let x = 4. 
A) 2x^2-16x+24 = 32 - 64 + 24 = -8 wrong answer B) -2x^2+16x-24 
So B must be the correct answer. 
SOLUTION 6 A) 2x^2-16x+24 = 2(x² - 8x + 12) = 2(x - 2)(x - 6) B) -2x^2+16x-24 = -2(x² - 8x + 12) = -2(x - 2)(x - 6) C) -2x^2-16x+24 = -2(x² + 8x - 12) doesn't factor D) -x^2-16x+12 = -(x² + 16x - 12) doesn't factor E) x^2-16x-24 doesn't factor 
The only choices that have x-intercepts of 2 and 6 are A and B. Since (4, 8) is a vertex, the quadratic expression must evaluate to 8 when x = 4 
A) 2x^2-16x+24 = 2(x² - 8x + 12) = 2(x - 2)(x - 6) = 2(2)(-2) = -8 wrong answer B) -2x^2+16x-24 = -2(x² - 8x + 12) = -2(x - 2)(x - 6) 
So B must be the solution