SOLUTION: Find the vertex, line of symmetry, the maximum or minimum value of the quadratic equation, and graph the function. f(x)= -2x^2+2x+6

Algebra ->  Points-lines-and-rays -> SOLUTION: Find the vertex, line of symmetry, the maximum or minimum value of the quadratic equation, and graph the function. f(x)= -2x^2+2x+6      Log On


   

Question 156941: Find the vertex, line of symmetry, the maximum or minimum value of the quadratic equation, and graph the function. f(x)= -2x^2+2x+6
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Find the vertex, line of symmetry, the maximum or minimum value of the quadratic equation, and graph the function. f(x)= -2x^2+2x+6
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set f(x) = 0 to find where it crosses the x-axis.
-2x^2+2x+6=0
x^2 - x - 3 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-1x%2B-3+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-1%29%5E2-4%2A1%2A-3=13.

Discriminant d=13 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--1%2B-sqrt%28+13+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-1%29%2Bsqrt%28+13+%29%29%2F2%5C1+=+2.30277563773199
x%5B2%5D+=+%28-%28-1%29-sqrt%28+13+%29%29%2F2%5C1+=+-1.30277563773199

Quadratic expression 1x%5E2%2B-1x%2B-3 can be factored:
1x%5E2%2B-1x%2B-3+=+%28x-2.30277563773199%29%2A%28x--1.30277563773199%29
Again, the answer is: 2.30277563773199, -1.30277563773199. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-1%2Ax%2B-3+%29

The vertex is at the minimum: set the 1st derivative to 0
2x-1=0
x = 1/2
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sub for x in the eqn
y = (1/4) - 1/2 - 3
y = -3 1/4 = -13/4
So the vertex is at (1/2,-13/4)
Since there's no xy term, the axis of symmetry is parallel to the y-axis. It goes thru the vertex, so it's:
x = 1/2