The equation of the quadratic function is y = -1/4(x + 2)² + 4
How to determine the quadratic equation?
A quadratic equation is represented as:
y = ax² +bx + c
When x = 0 and y = 3, we have:
3 = a(0)² +b(0) + c
This gives
c = 3
So, we have:
y = ax² +bx + 3
When x = 1 and y = 1.75, we have:
1.75 = a(1)² +b(1) + 3
This gives
a + b = -1.25
Make b the subject
b = -1.25 - a
When x = -2 and y = 4, we have:
4 = a(-2)² +b(-2) + 3
This gives
4a - 2b = 1
Substitute b = -1.25 - a in 4a - 2b = 1
4a - 2(-1.25 - a) = 1
Expand
4a + 2.5 + 2a = 1
Evaluate the like terms
6a = -1.5
Divide both sides by 6
a = -1/4
Substitute a = -1/4 in b = -1.25 - a
b = -1.25 + 1/4
Evaluate
b = -1
So, we have:
y = -1/4x² - x + 3
Factor out -1/4
y = -1/4(x² + 4x) + 3
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Take the coefficient of x
k = 4
Divide by 2
k/2 = 2
Square both sides
(k/2)² = 4
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Add and subtract 4 inside the bracket of y = -1/4(x² + 4x) + 3
y = -1/4(x² + 4x + 4 - 4) + 3
Expand
y = -1/4(x² + 4x + 4) + 1 + 3
Evaluate the sum
y = -1/4(x² + 4x + 4) + 4
Express the bracket as a perfect square
y = -1/4(x + 2)² + 4
Hence, the equation of the quadratic function is y = -1/4(x + 2)² + 4
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