Graph square root of 4-x^2

Problem

y=√(,4−x2)

Solution

1. Identify the function type as the upper half of a circle. The equation y=√(,4−x2) is derived from x2+y2=4 where y≥0

2. Determine the domain by setting the radicand to be greater than or equal to zero: 4−x2≥0

3. Solve the inequality to find the interval for x

x2≤4

−2≤x≤2

4. Find the range by evaluating the function at the boundaries and the vertex. Since the square root is non-negative and the maximum value of 4−x2 is 4 the range is 0≤y≤2

5. Identify the intercepts. The x-intercepts occur at (−2,0) and (2,0) and the y-intercept occurs at (0,2)

6. Sketch the graph as a semi-circle centered at the origin (0,0) with a radius of 2 plotted only for y≥0

Final Answer

y=√(,4−x2)

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