This is a polynomial function called Quadratic Function. This is written as:
\(f(x)=(x+1)^2+2\)
Recall that the graph of a quadratic function is a special type of U-shaped curve called parabola. The graph of this function is shown in the Figure below. From that, we can say the following:
A. The domain is all real numbers.
This is true. In fact, the domain of any polynomial function is the set of all real numbers.
B. The range is all real numbers greater than or equal to 1.
This is false. Instead, the range is all real numbers greater than or equal to 2, that is the \(y-coordinate\) of the vertex.
C. The y-intercept is 3.
This is true. We can find the \(y-intercept\) letting \(x=0\), so:
\(x=0 \\ \\ y-intercept \rightarrow f(0)=(0+1)^2+2=3\)
D. The graph of the function is 1 unit up and 2 units to the left from the graph of \(y=x^2\)
This is false. The correct statement is that the graph of the function is 2 units up and 1 unit to the left from the graph of \(y=x^2\)
E. The graph has two x-intercepts.
This is false. As shown the graph below, this function does not have any \(x-intercept\)
