Question 442317: Please help me with this problem. Find the vertex, the line of symmetry, the maximum or minimum value of the quadratic function, and graph the function.
f(x)= -2x^2+2x+7
Answer by swincher4391(1107)   (Show Source):
You can put this solution on YOUR website! If you recall, the vertex of a parabola is defined as (-b/2a , f(-b/2a) where your problem is in the form ax^2+bx+c.
So, given that:
-b/2a = -2 / -4 = 1/2.
So what's f(1/2)? -2(1/2)^2 + 2(1/2) + 7 = -1/2 + 1 + 7 = 8 - 1/2 = 7 1/2 = 15/2.
So our vertex is (1/2, 15/2).
Since the parabola is symmetrical vertically, we need a vertical line to satisfy our line of symmetry. This means we need to take a line of the form x= ?. By definition, the vertex is the center-most point on the quadratic. So, we can take the x-coordinate of the vertex as our line of symmetry. So our line of symmetry is  .
The maximum or minimum value is determined by the y-value, and by definition, the y-coordinate of the vertex is going to be a maximum or minimum. The question is, "which one is it?" Since the coefficient of your x^2 term is negative, then the quadratic will be opening down. This means the vertex of your quadratic will be a MAXIMUM. The maximum value of your function is 15/2.
Luckily, we have a built in graphing utility, and I will use it, but let's say we didn't.
Though it may not be the most accurate, you're not always going to have a calculator. The things you need to know to graph a quadratic are:
1) the vertex... well what do you know we have that.
2) the roots (x-intercepts) of the quadratic
Let's find the roots of this quadratic.
This won't be pretty, but we'll have to use the quadratic formula to come up with the roots.
Recall that the equation for the quadratic formula is  .
So -2 +- sqrt(4 + 56) / -4.
That's UGLY!
we can guess that sqrt(4+56) =sqrt(60)... since sqrt(60) < sqrt(64). We can say that this number is going to be slightly under 8.
We'll call it 7.8 as a total guess.
-2+7.8 / -4 = 5.8/-4 = -1.4 ish?
-2-7.8/-4 = -9.8/-4 = 2.4 ish?
So you would plot these points, (2.4,0) , (-1.4,0) and (1/2, 15/2). Draw the quadratic the best you can.
Now here are the real roots and graph:
| Solved by pluggable solver: SOLVE quadratic equation with variable |
Quadratic equation  (in our case  ) has the following solutons:
For these solutions to exist, the discriminant  should not be a negative number.
First, we need to compute the discriminant :  .
Discriminant d=60 is greater than zero. That means that there are two solutions:  .
Quadratic expression  can be factored:
Again, the answer is: -1.43649167310371, 2.43649167310371.
Here's your graph:
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You'll notice my rough estimates would have gotten the job done for a rough sketch. It is essential to be able to estimate like this when you don't have a calculator.
I hope this helped!
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