Extreme value of quadratic equation?

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Hoodleehoo

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I'm having trouble understanding this. The question is about a function f(x) which equals a quadratic equation. It is asking what the minimum value the output would be. The answer states that "to find the extreme value of a quadratic equation, find the number exactly between the two possible values for x" (with it equal to zero, obviously)

It doesn't explain this any further. I get that to get the right answer I have to solve the quad equation for zero and get the two values for x. But I don't understand what it means by "extreme value".
 
Here, "extreme value" is just a stand-in for saying "minimum or maximum."
 
Hoodleehoo said:
The answer states that "to find the extreme value of a quadratic equation, find the number exactly between the two possible values for x"

It doesn't explain this any further.
Click to expand...

If that's all the explanation says, then it's incomplete.

The value of x halfway between the x-intercepts only tells you where the minimum occurs. It is not the minimum function value itself.

Once you determine this value of x, you then need to evaluate f(x), to obtain the function's minimum value.

By the way, there's another method for finding that value of x -- IF you know the coefficients a and b (that is, if you know the values of a,b in the quadratic polynomial ax^2+bx+c that defines the function).

The minimum (or maximum) value of a quadratic function always occurs at the value of x given by this formula:

x = -b/(2a)


Here's an example:

g(x) = 3x^2 - 7x + 10, find the minimum value of function g

Using the formula x=-b/(2a) to find the location, we get x = 7/6

Then, evaluating g(7/6) we get 71/12.

The minimum value of function g is 71/12. :)
 
Okay so that tells me what to use to get the minimum value. Why is it that halfway between the numbers results in the minimum value? Any clever way of describing that to a person who is a real dunce at math? Lol
 
Hoodleehoo said:
Why [does the local extreme value on a parabola occur] halfway between the [x-intercepts]?
Click to expand...

This happens because the shape of a parabola is symmetric. That is, if you draw a vertical line through the minimum (or maximum) point on the graph of y=ax^2+bx+c, the left half is a mirror image of the right half. We call this line the "axis of symmetry".

Therefore, if you need to move, say, six units to the right of the axis of symmetry to reach an x-intercept, you will also need to move six units to the left of the axis of symmetry to reach the other x-intercept. So, the local extreme value is halfway inbetween the x-intercepts.

Ya know, you're not a dunce. I have not yet learned to play the piano very well at all; does that mean I'm a dunce? ;) (Don't answer that, Jomo.)
 
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That's a great explanation. Is there a diagram of this anywhere that I can see? Of this specific type of curve with labeled X intercepts I mean.
 
Hoodleehoo said:
That's a great explanation. Is there a diagram of this anywhere that I can see? Of this specific type of curve with labeled X intercepts I mean.
Click to expand...
Seriously you can draw your own diagram. As mentioned just make sure that you start with a vertical line (draw it lightly as it is not part of the graph), then pick a point on the line for the minimum and then draw your 'U' shape curve. Now draw any (actually draw a few) horizontal line that crosses the the graph in two places and use that line as the x-axis. Now to you see that the x value for that minimum point is right in the middle of the two x intercepts?
 
Otis said:
This happens because the shape of a parabola is symmetric. That is, if you draw a vertical line through the minimum (or maximum) point on the graph of y=ax^2+bx+c, the left half is a mirror image of the right half. We call this line the "axis of symmetry".

Therefore, if you need to move, say, six units to the right of the axis of symmetry to reach an x-intercept, you will also need to move six units to the left of the axis of symmetry to reach the other x-intercept. So, the local extreme value is halfway inbetween the x-intercepts.

Ya know, you're not a dunce. I have not yet learned to play the piano; does that mean I'm a dunce? ;) (Don't answer that, Jomo.)
Click to expand...
What did I do??? Should I go to the corner again for something I did not do? (btw it was Denis who did it)
 
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