Graph f ( x ) = - x2 + 2x + 3 .
Solution
It is a parabola in the general form.
f ( x ) = ax2 + bx + c
In this form, the x-coordinate of the vertex (on the parabola the highest or lowest point) is
x = - b/2a and we get the y-coordinate is y = f ( - b/2a) . Hence, for our parabola the coordinates of the vertex will be.
x= - 2/(2(-1)=1
y=f(1) = -(1) 2 + 2(1) +3=4
Hence, the vertex for this parabola is (1,4).
We can also find out which direction the parabola opens from the sign of a. If a is +ve the parabola opens up & if a is -ve the parabola opens down. In our case the parabola opens down.
Now, since the vertex is above the x-axis & the parabola opens down we know that we'll have x-intercepts (that means values of x for which we'll have f ( x ) = 0 ) on this graph. Hence, we'll solve out the following.
- x2 + 2 x + 3 = 0
x2 - 2 x - 3 = 0
( x - 3) ( x + 1) = 0
Hence, we will have x-intercepts at x = -1 and x = 3 . Notice as well that to make our life simple in the solution process we multiplied everything by -1 to get the coefficient of the x2 positive. It made the factoring easier.
Following is a sketch of this parabola.
