Graph y=2-x^2

Problem

y=2−x2

Solution

1. Identify the type of function. The equation y=2−x2 is a quadratic function in the form y=a*x2+b*x+c which represents a parabola.

2. Determine the orientation. Since the coefficient of x2 is a=−1 the parabola opens downward.

3. Find the vertex. The vertex occurs at x=(−b)/(2*a) Here b=0 so x=0 Substituting x=0 into the equation gives y=2−0=2 The vertex is (0,2)

4. Calculate the y-intercept. Setting x=0 results in y=2 The y-intercept is (0,2)

5. Calculate the x-intercepts. Set y=0 and solve 0=2−x2

x2=2

x=±√(,2)≈±1.41

6. Plot additional points to define the shape. For x=1 y=2−1=1 For x=2 y=2−2=−2 Due to symmetry across the y-axis (x=0, the points (−1,1) and (−2,−2) are also on the graph.

7. Sketch the curve. Draw a smooth, downward-opening curve passing through the vertex (0,2) the x-intercepts (±√(,2),0) and the points (±1,1) and (±2,−2)

Final Answer

y=2−x2

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