Question 186455: write the expression 9x^2-9x+1 in the form a(x+b)+c where a, b, and c are real numbers.
Hence, state whether the function y = 9x^2-9x+1 has a maximum or minimum value.
state the value of x at which this maximum or minimum value occurs
Answer by feliz1965(151)   (Show Source):
You can put this solution on YOUR website! To write 9x^2 - 9x + 1 in the form a(x + b) + c, we need to concentrate on the first-two terms of the given expression.
The terms 9x^2 - 9x can be factored to be 9x(x - 1) and then we add the given 1.
So, 9x^2 - 9x + 1 becomes 9x(x - 1) + 1.
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In the given equation, y = 9x^2- 9x + 1, we see that the coefficient of the x^2 term is a positive number. Do you see positive 9? When the coefficient of the x^2 term is positive, the parabola will open upward indicating minimum value and when the same coefficient is negative, the parabola will open downward indicating maximum value. This parabola has minimum value based on the fact that the coefficient of the x^2 term is positive.
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To find the vertex value of x for this parabola, we can use the formula
x = -b/2a.
In the given equation, a = 9 and b = -9.
We plug and chug.
x = -(-9)/2(9)
x = 9/18
x = 1/2
To find the y value of the vertex, we plug 1/2 into the given equation and simplify.
y = 9(1/2)^2 -9(1/2) + 1
y = -(5/4)
The vertex is the point (1/2, -5/4).
The value of x where the given parabola will have minimum value is x = 1/2.
NOTE: I know you did not request the vertex but I just thought you may want to know how it is found.
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