Graph f(x)=-x^3-1

Problem

ƒ(x)=−x3−1

Solution

1. Identify the parent function and its transformations. The base function is y=x3 The negative sign indicates a reflection across the xaxis, and the −1 indicates a vertical shift downward by 1 unit.

2. Determine the y-intercept by evaluating the function at x=0

ƒ(0)=−(0)3−1

ƒ(0)=−1

The y-intercept is (0,−1)

3. Determine the x-intercept by setting ƒ(x)=0 and solving for x

−x3−1=0

−x3=1

x3=−1

x=−1

The x-intercept is (−1,0)

4. Calculate additional points to define the shape of the curve.
   For x=1

ƒ(1)=−(1)3−1=−2

For x=−2

ƒ*(−2)=−(−2)3−1=8−1=7

5. Sketch the graph by plotting the points (−2,7) (−1,0) (0,−1) and (1,−2) The curve starts in the second quadrant, passes through the intercepts with a flat inflection point at (0,−1) and continues downward into the fourth quadrant.

Final Answer

ƒ(x)=−x3−1

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