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: _________
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. 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 x-coordinate of the axis of symmetry, which is 2, into the function to get f(2) = -2(2)^2 + 8(2) + 10 = -8 + 16 + 10 = 18. Therefore, the vertex is (2, 18).
Since the coefficient of the x^2 term is negative, the parabola is concave down and the vertex represents the maximum point. So, the maximum value is 18.
The domain of a quadratic function is always all real numbers, so the domain in this case is (-∞, ∞).
To find the range, we can consider that since the coefficient of the x^2 term is negative, the parabola opens downward and its highest point is the vertex. Therefore, the range is (-∞, 18].
To find the x-intercepts, we set f(x) = 0 and solve for x. In this case, we have -2x^2 + 8x + 10 = 0. Using factoring or the quadratic formula, we find the solutions to be x = -1 and x = 5. So, the x-intercepts are -1 and 5.
To find the y-intercept, we set x = 0 in the equation of the function: f(0) = -2(0)^2 + 8(0) + 10 = 0 + 0 + 10 = 10. Therefore, the y-intercept is (0, 10).
To summarize:
Axis of symmetry: x = 2
Vertex: (2, 18)
Minimum or Maximum: Maximum (at the vertex)
Domain: (-∞, ∞)
Range: (-∞, 18]
x-intercepts/zeros: -1, 5
y-intercept: (0, 10)