SOLUTION: please help me sketch the graph for {{{-x^2+ 2x +3}}} and what is the range ,please show all working.

Question 1117498: please help me sketch the graph for and what is the range ,please show all working.
Found 2 solutions by rothauserc, greenestamps:
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
f(x) = -x^2 +2x +3
:
this is a parabola that curves downward
:
the x coordinate for the vertex is -b/2a = -2/2(-1) = 1
:
f(1) is the y coordinate(highest point of the parabola) = -(1^2) +2(1) +3 = 4
:

:
range of f(x) over the real numbers is f(x) < or = 4
:

Answer by greenestamps(13215) (Show Source): You can put this solution on YOUR website!

The solution by the other tutor, using x = -b/2a for the x coordinate of the vertex, is one good way to work a problem like this.

Another useful method to know is to put the equation in standard form:

or

In either of these forms, the vertex is at (h,k).

For your example....

factor out the leading coefficient

complete the square

or

These forms tell you that the vertex is at (1,4); the a=-1 tells you that the parabola opens downward. That makes the maximum value of the function is 4; so the range is y <= 4.
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