Graph f(x)=x^2-6

Problem

ƒ(x)=x2−6

Solution

1. Identify the type of function. This is a quadratic function in the form ƒ(x)=a*x2+b*x+c where a=1 b=0 and c=−6 The graph is a parabola that opens upward because a>0

2. Determine the vertex. Since b=0 the xcoordinate of the vertex is x=−b/(2*a)=0 Substituting x=0 into the function gives ƒ(0)=−6 The vertex is (0,−6)

3. Find the yintercept. Set x=0 to find the point where the graph crosses the yaxis.

ƒ(0)=0−6=−6

4. Find the xintercepts. Set ƒ(x)=0 and solve for x

x2−6=0

x2=6

x=±√(,6)≈±2.45

5. Plot additional points to define the shape. For x=1 ƒ(1)=1−6=−5 For x=2 ƒ(2)=2−6=−2 For x=3 ƒ(3)=3−6=3 Use symmetry across the yaxis to find the corresponding points (−1,−5) (−2,−2) and (−3,3)

6. Sketch the curve. Draw a smooth, U-shaped curve passing through the vertex (0,−6) the xintercepts (±√(,6),0) and the calculated points.

Final Answer

ƒ(x)=x2−6

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