Graph the function f(x) = -2x^2 + 8x + 10 and identify the the attributes.
Axis of symmetry:
Vertex:
Minimum of Maximum?:
Domain: _________ and Range: ________
x-intercepts/zeros: ________ __________
y-intercept: _________
Options are Minimum, (2,0), Maximum, (2,18), (10,0), (18,2), x = 2, (-1,0), (-5,0), (0,10) y ≤ 18, None, y ≤ 2, All real numbers, x = 18, (5,0)
To graph the function f(x) = -2x^2 + 8x + 10, we can start by finding the axis of symmetry using the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic equation. In this case, a = -2 and b = 8, so the axis of symmetry is x = -8/(2*(-2)) = -8/(-4) = 2.
To find the vertex, we substitute the value of x = 2 into the equation to find the corresponding y-coordinate:
f(2) = -2(2)^2 + 8(2) + 10 = -8 + 16 + 10 = 18
Therefore, the vertex is (2, 18).
Since the coefficient of x^2 is negative, the parabola opens downwards, indicating that the vertex is a maximum point. So, the maximum of the function is at the vertex, which is (2, 18).
The domain of the function is all real numbers unless specifically restricted. In this case, there are no restrictions, so the domain is all real numbers: (-∞, ∞). The range can be determined by looking at the graph, and we can see that the parabola opens downwards and the vertex is at (2, 18). Therefore, the range is y ≤ 18.
To find the x-intercepts or zeros of the function, we set f(x) = 0 and solve for x:
-2x^2 + 8x + 10 = 0
Dividing through by -2, we get:
x^2 - 4x - 5 = 0
Factoring, we have:
(x - 5)(x + 1) = 0
So, the x-intercepts or zeros are x = 5 and x = -1.
To find the y-intercept, we set x = 0 in the function:
f(0) = -2(0)^2 + 8(0) + 10 = 0 + 0 + 10 = 10
Therefore, the y-intercept is (0, 10).
The attributes of the function are as follows:
Axis of symmetry: x = 2
Vertex: (2, 18)
Minimum of Maximum: Maximum
Domain: All real numbers (-∞, ∞)
Range: y ≤ 18
x-intercepts/zeros: (5, 0) and (-1, 0)
y-intercept: (0, 10)