Question

How do you find the vertex and intercepts for `f(x) = 10x^2 - 5x + 12`?

Answer 1

Vertex (`1/4, 91/8)`
No x-intercepts

Explanation:

`f(x) = 10x^2 - 5x + 12`
x - coordinate of vertex:
`x = -b/(2a) = 5/20 = 1/4`
y-coordinate of vertex:
`f(1/4) = 10(1/16)- 5(1/4) + 12 = 5/8 - 10/8 + 96/8 = 91/8`
Vertex `(1/4, 91/8)`
y-intercept when x = 0, --> f(0) = 12
x- intercepts when f(x) = 0. Solve the quadratic equation:
`f(x) = 10x^2 - 5x + 12 = 0`
`D = d^2 = b^2 - 4ac = 25 - 480 = -455 < 0.`
There are no x-intercepts. The graph of f(x) is an upward parabola (a > 0), that completely stays above the x-axis.