Calculate the Positive Area of y=(x-4)²+1: Shifted Parabola Problem

Find the positive area of the function

y=(x4)2+1 y=(x-4)^2+1

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1

Understand the problem

Find the positive area of the function

y=(x4)2+1 y=(x-4)^2+1

2

Step-by-step solution

To solve this problem, we will analyze the given quadratic function:

The function is y=(x4)2+1 y = (x-4)^2 + 1 .

Step 1: Identify the vertex of the parabola.
The function y=(x4)2+1 y = (x-4)^2 + 1 is in the form y=(xh)2+k y = (x-h)^2 + k , where h=4 h = 4 and k=1 k = 1 . This gives the vertex at the point (4,1) (4, 1) .

Step 2: Analyze the shape and direction of the parabola.
This parabola opens upwards because the coefficient of (x4)2 (x-4)^2 is positive. Hence, the vertex is the minimum point of the parabola.

Step 3: Determine when the function is positive.
Since the minimum value of the function at the vertex is 1 1 , and all quadratic functions in the form of (xh)2+k (x-h)^2 + k are non-negative, this means the function y=(x4)2+1 y = (x-4)^2 + 1 is always positive for any real number x x .

Step 4: Comparison with the choices:
From the explanation, the function is always positive for all x x . Thus, the correct choice is: For all X.

The positive area of the function covers all real numbers x x .

Therefore, the function is positive for all x x .

3

Final Answer

For all X

Practice Quiz

Test your knowledge with interactive questions

Which equation represents the function:

\( y=x^2 \)

moved 2 spaces to the right

and 5 spaces upwards.

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