Identify any intercepts, relative extrema, points of inflection, and asymptotes.
y = 4x^2 − 24x + 34
X-intercept ? (x, y) (smaller x-value)
(x, y) (larger x-value)
relative minimum (x, y)
To find the x-intercepts, we set y = 0 and solve for x. So we need to solve the equation 4x^2 - 24x + 34 = 0.
The discriminant of this quadratic equation is b^2 - 4ac, where a = 4, b = -24, and c = 34.
So the discriminant is (-24)^2 - 4(4)(34) = 576 - 544 = 32.
Since the discriminant is positive, there are two distinct real solutions for x, and therefore two x-intercepts.
To find the x-values of the x-intercepts, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
Plugging in the values for a, b, and c, we get x = (-(-24) ± √(32)) / (2 * 4).
Simplifying further, we have x = (24 ± √32) / 8.
This gives us two possible x-values for the intercepts.