Final answer:
a) The function is increasing for x > -2. b) The relative minimum of the graph occurs at x = -2. c) The number where the graph has a relative minimum is -2. d) The domain of the function is all real numbers. e) The range of the function is y ≥ -1.
Step-by-step explanation:
a) To identify the interval in which the function is increasing, we need to find where the derivative of the function is positive. Taking the derivative of the given function, we get:
y' = 2x + 4
Setting y' greater than 0 and solving for x:
2x + 4 > 0
x > -2
So, the function is increasing for x > -2.
b) To find the relative minimum of the graph, we need to find the critical points. Taking the second derivative of the function:
y'' = 2
The second derivative is positive, which means the function has a relative minimum. To find the x-coordinate of the relative minimum, we set the first derivative equal to zero:
2x + 4 = 0
x = -2
So, the x-coordinate of the relative minimum is -2.
c) The number where the graph has a relative minimum is -2.
d) The domain of the function is all real numbers because there are no restrictions on the value of x.
e) The range of the function can be determined by looking at the vertex of the graph. The x-coordinate of the vertex is -2, and substituting this value into the original function, we get:
y = (-2)² + 4(-2) + 3 = -1
So, the range of the function is y ≥ -1.