Its been awhile since I have done compleating the square, can you please help me?
Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point.
19.f(x) = x2 - 2x + 1
And how you would complete the square when I have f(x) = 2x2 - 4x
completing the square
2(x^2 -2x +1 -1)
2(x-1)^2 -2
19.
f'=2x-2 set that to zero. solve for x.
Now use the second derivative test...
f"=2, positive, so the conclusion is that it is ....
I will be happy to critique your thinking
2x2-16x+4 = i do not know
To determine whether a quadratic function has a minimum or maximum value, you can look at the coefficient of the x² term. If the coefficient is positive, like in the first function f(x) = x² - 2x + 1, the quadratic will have a minimum value. If the coefficient is negative, the quadratic will have a maximum value.
For the given function f(x) = x² - 2x + 1:
Step 1: Notice that the coefficient of x² is 1, which is positive. So, the function has a minimum value.
Step 2: To find the coordinates of the minimum point, we need to complete the square.
Step 3: Start by taking half of the coefficient of x, which is -2/2 = -1.
Step 4: Square this value: (-1)² = 1.
Step 5: Add this value inside the parentheses to the equation, and also subtract it outside the parentheses to keep the equation balanced:
f(x) = (x² - 2x + 1) + 1 - 1
= (x - 1)² + 1 - 1
= (x - 1)²
From the completed square form, we can see that the minimum point is at (1,0). The minimum value of f(x) = x² - 2x + 1 is 0.
Now, let's move on to the second question: How to complete the square when you have f(x) = 2x² - 4x?
For the given function f(x) = 2x² - 4x:
Step 1: Factor out the common factor, which is 2:
f(x) = 2(x² - 2x)
To complete the square, we only need to focus on the quadratic part inside the parentheses, which is x² - 2x.
Step 2: Take half of the coefficient of x, which is -2/2 = -1.
Step 3: Square this value: (-1)² = 1.
Step 4: Add this value inside the parentheses to the equation, and also subtract it outside the parentheses to keep the equation balanced:
f(x) = 2(x² - 2x + 1) - 2
= 2(x - 1)² - 2.
The completed square form of f(x) = 2x² - 4x is 2(x - 1)² - 2.