Question

Y=-0.5x^2+2x+16.
y-intercept?
X-intercept(s)?
Coordinates of the vertex?
Equation of axis of symmetry?
Maximum or minimum?

Answer 1

To find the y-intercept, set x to zero and solve for y:

y = -0.5(0)^2 + 2(0) + 16
y = 16

So the y-intercept is at (0, 16).

To find the x-intercepts, set y to zero and solve for x:

0 = -0.5x^2 + 2x + 16
Multiplying by -2 to simplify:
0 = x^2 - 4x - 32

This equation can be factored as:

0 = (x - 8)(x + 4)

Thus, x-intercepts are at x = 8 and x = -4.

To find the coordinates of the vertex, use the formula x = -b/(2a) to find the x-coordinate of the vertex:

x = -2 / (2*(-0.5))
x = -2 / -1
x = 2

Now substitute x = 2 into the original equation to find the y-coordinate of the vertex:

y = -0.5(2)^2 + 2(2) + 16
y = -2 + 4 + 16
y = 18

So the vertex is at (2, 18).

The equation of the axis of symmetry is x = 2.

Since the coefficient of the x^2 term is negative, the parabola opens downwards, making the vertex a maximum point. Thus, the vertex represents the maximum value of the function.