Given: y = 8(x-3)^2 - 5
color(blue)("Find the vertex:")
From the vertex form: y = a(x - h)^2 + k, where vertex (h, k)
we can find the vertex of the given equation as (3, -5)
color(blue)("Find the "y-"intercept") by setting x = 0:
y = 8(0-3)^2 - 5
y = 8*9 - 5 = 72 - 5 = 67
y-intercept: (0, 67)
color(blue)("Find the "x-"intercepts") by setting y = 0:
0 = 8(x-3)^2 - 5
Distribute using the square function: (a + b)^2 = a^2 + 2ab + b^2
8 (x^2 - 6x + 9) - 5 = 0
8x^2 - 48x + 72 - 5 = 0
8x^2 - 48x + 67 = 0
Use the quadratic formula to solve:
x =( -B +- sqrt(B^2 - 4AC))/(2A),
where the equation is in the form: Ax^2 + Bx + C = 0
x =( 48 +- sqrt((-48)^2 - 4*8*67))/(16)
x = 48/16 +- sqrt(160)/16
x = 3 +- (sqrt(16 * 10))/16
x = 3 +- (4 sqrt(10))/16
x = 3 +- sqrt(10)/4
x-intercepts: (3 - sqrt(10)/4, 0), (3 + sqrt(10)/4, 0)
