Answer:
a. Parabola opens upward
b. Vertex at (4, 7)
c. No x-intercept
d. y-intercept = 39 at point(0, 39)
e. Attached
Explanation:
The function is a quadratic function and the function graph represents a parabola
The equation of a parabola in vertex form is
y = a(x -h)² + k
The vertex is found at (h, k)
If a is positive, the parabola opens upward, if negative, the parabola opens downward.
Mapping the given function f(x) = y = 2(x - 4)² + 7 to the general equation we can see that
a = 2
h = 4
k = 7
a. Since a >0 the parabola opens downward
b. The vertex is at (h, k) => (4, 7)
c. The x-intercept can be found by setting y = 0 and solving for x
Setting y = 0 gives
0 = 2(x -4)² + 7
Switching sides,
2(x-4)² + 7 = 0
2(x-4)² = -7
(x-4)² = -7/2
(x - 4) =√(-7/2)
Since square roots of negative numbers are not real numbers, the parabola does not have an x-intercept
c. To find the y-intercept set x = 0 and solve for y
=> y = 2(0-4)² + 7
y = 2 (-4)² + 7
y = 2 x 16 + 7
y = 39
So y-intercept is at the point(0, 39)
e. Graph provided