Graph f(x)=x^3-5

Problem

ƒ(x)=x3−5

Solution

1. Identify the parent function as y=x3 which is a cubic function with a characteristic "S" shape passing through the origin (0,0)

2. Determine the transformation by observing the constant −5 subtracted from the function.

3. Apply a vertical shift downward by 5 units to the parent function.

4. Calculate key points to plot the graph:

   • When x=−1 ƒ*(−1)=(−1)3−5=−6

   • When x=0 ƒ(0)=(0)3−5=−5 (the y-intercept)

   • When x=1 ƒ(1)=(1)3−5=−4

   • When x=2 ƒ(2)=(2)3−5=3

5. Find the x-intercept by setting ƒ(x)=0

x3−5=0

x3=5

x=√(3,5)≈1.71

6. Sketch the curve through the points (−1,−6) (0,−5) (1,−4) and (2,3) ensuring the graph continues infinitely in both directions.

Final Answer

ƒ(x)=x3−5

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