Answer:
Step-by-step explanation:
f(x) = −16x2 + 24x + 16 = 0 can be reduced to -2x^2 + 3x + 2 = 0 and then to 2x^2 - 3x - 2 = 0. Solve this for x using the quadratic formula:
The discriminant is b^2 - 4ac = (-3)^2 - 4(2)(-2) = 9 + 16 = 25.
Therefore the roots are:
-(-3) ± √25 3 ± 5
x = ----------------- = ------------ => x = 2 and x = -1/2
2(2) 4
The x-intercepts are points: (2, 0) and (-1/2, 0)
Because the coefficient of the x^2 term is negative, this graph opens down and the vertex represents a maximum.
To graph this function, find and plot the vertex. It is exactly halfway between the x-intercepts, that is, at x = 1 1/4, for which the y value is 21:
vertex and maximum at (5/4, 21)
Finally, find the y-intercept. Let x = 0; we find that y = 16. The y-intercept is 0, 16)
We now have four points on the graph and know where the maximum is. Plot this max (5/4, 21) and the x-intercepts (2, 0) and (-1/2, 0), and finally the y-intercept. Draw a smooth curve through these points, remembering that the graph is symmetrical about the axis of symmetry x = 5/4.
