3(x - 6) + 6 = 5x - 6. what is the solution
2 answers:
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Answer: The <u> maximum </u> value is <u> -3.75 </u>
Note: -3.75 = -15/4
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Explanation:
The given quadratic is in the form y = ax^2 + bx + c, where,
- a = -1
- b = 1
- c = -4
We'll plug the values of 'a' and b into the formula below
h = -b/(2a)
h = -1/(2(-1))
h = 1/2
h = 0.5
This represents the x coordinate of the vertex. Recall the vertex is (h,k).
Plug this x value into the function to find its corresponding y coordinate
y = -x^2 + x - 4
y = -(0.5)^2 + 0.5 - 4
y = -3.75
The vertex is located at (0.5, -3.75)
Since a < 0, this means the parabola opens downward and that the vertex is the highest point on the parabola. Therefore, the largest y can get is y = -3.75 which we consider the maximum.
The graph visually helps confirm this (see image attachment).
Side note: -3.75 = -15/4
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Here's another way we can find the vertex (h,k). We'll complete the square to get the function into vertex form
y = -x^2 + x - 4
y + 4 = -x^2 + x
y + 4 = -(x^2 - x)
y + 4 = -(x^2 - x + 0)
y + 4 = -(x^2 - x + 0.25 - 0.25) .... see note below
y + 4 = -(x^2 - x + 0.25) -(-0.25)
y + 4 = -(x^2 - x + 0.25) + 0.25
y + 4 = -(x - 0.5)^2 + 0.25
y = -(x - 0.5)^2 + 0.25 - 4
y = -(x - 0.5)^2 - 3.75
This is in the form y = a(x-h)^2 + k where (h,k) = (0.5, -3.75).
Note: The x coefficient is -1 which cuts in half to -0.5, then that squares to 0.25
