Which of these graphs most resembles the graph of
𝑦
=
(
𝑥
−
4
)
3
at
𝑥
=
3
?

Question

Which of these graphs most resembles the graph of 
𝑦
=
(
𝑥
−
4
)
3
 at 
𝑥
=
3
?

BlackTom AI · Accepted Answer

The provided data for the question is incomplete: there are no answer options given, and the answer field is empty. To determine which graph best resembles the function at x = 3, we can still outline the correct reasoning you would use if the options were present.

First, rewrite the function in a familiar form: y = (x − 4)^3. This is a cubic function that has been shifted to the right by 4 units compared to the basic y = x^3.

Key characteristics to examine when comparing graphs:
- The point at x = 3: plug in to find the y-coordinate, y(3) = (3 − 4)^3 = (−1)^3 = −1. So the graph passes through the point (3, −1).
- The slope at x = 3: compute the derivative y' = 3(x − 4)^2. Evaluating at x = 3 gives y'(3) = 3(−1)^2 = 3. So the tangent line there has slope 3, which is relatively steep and positive.
- Inflection point: a cubic y = (x − h)^3 has its inflection point at x = h, i.e., here at x = 4. This tells you the graph is increasing throughout, with an S-shape centered around x = 4.

When comparing candidate graphs (had they been provided):
- Look for a cubic-shaped curve that passes through (3, −1).
- Check that the slope of the curve at x = 3 is about 3 (the tangent line there should be fairly steep and rising to the right).
- Ensure the inflection point is near x = 4, consistent with a rightward shift of the standard y = x^3.

Without the actual answer options, you cannot select the correct graph, but these are the precise features you would use to evaluate any given graphs against the function y = (x − 4)^3 at x = 3.

MATH-112-301-001 Unproctored Midcourse Exam 3 Practice Exam 1

Which of these graphs most resembles the graph of 𝑦 = ( 𝑥 − 4 ) 3 at 𝑥 = 3 ?

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