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Chapter 3: Problem 84

Determine if the statement is true or false. The graph of \(f(x)=3 x^{2}(x-4)^{4}\) has no points \(\mathrm{in}\) quadrants III or IV.

Short Answer

Expert verified
True. The graph has no points in quadrants III or IV.

Step by step solution

01

Identify the function and its properties

The given function is: \[ f(x) = 3x^2(x - 4)^4 \]Since the highest power of the polynomial is even, the function will tend towards positive infinity as \(x\) tends towards positive or negative infinity.
02

Analyze the factors of the function

The function can be factored as \( 3x^2 \) and \( (x - 4)^4 \). Both of these factors are even powers, which means that they are always non-negative for any real number \(x\).
03

Consider the sign of the function

Both \( 3x^2 \) and \( (x - 4)^4 \) are non-negative for all \(x\), which implies that \( f(x) \) is always non-negative since it is the product of two non-negative terms.
04

Determine the regions where the function is positive or zero

\( f(x) = 0 \) at \( x = 0 \) or \( x = 4 \), which indicates the points where the function touches or crosses the X-axis. For all other values of \(x\), \( f(x) > 0 \).
05

Analyze the function's behavior in quadrants III and IV

In quadrant III, \(x < 0\) and \(y < 0\). In quadrant IV, \(x > 0\) and \(y < 0\). Since \( f(x) \) is never negative, it cannot exist in quadrants III or IV.
06

Conclusion

Since \(f(x) \) is never negative and only zero or positive, the function does not have any points in quadrants III (where both \(x\) and \(y\) are negative) or IV (where \(x > 0\) and \(y < 0\)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

graph behavior
In this section, we examine how the graph of a polynomial behaves. For the given function \( f(x) = 3x^2(x - 4)^4 \), let's understand the behavior at different values of \( x \).

First, note that \( 3x^2 \) and \( (x-4)^4 \) are both non-negative (they do not produce negative values), as squares and fourth powers are always non-negative. This means that the value of \( f(x) \) will always be non-negative as well.

As \( x \) moves towards positive infinity (\
quadrants
When analyzing where the graph of \( f(x) = 3x^2(x - 4)^4 \) is located, we observe four quadrants on the Cartesian plane:
  • Quadrant I: where both \( x \) and \( y \) are positive.
  • Quadrant II: where \( x \) is negative and \( y \) is positive.
  • Quadrant III: where both \( x \) and \( y \) are negative.
  • Quadrant IV: where \( x \) is positive and \( y \) is negative.
Since \( f(x) \) is always non-negative, it only exists in the upper two quadrants:
  • Quadrant I: when \( x \) > 4. \( f(x) \) is positive because all terms are positive or zero.
  • Quadrant II: when \( x \) < 0. \( f(x) \) is still non-negative (positive or zero) because both terms are transformed to positive by squaring.
Therefore, the graph does not exist in Quadrants III (\( x < 0 \) and \( y < 0 \)) or IV (\( x > 0 \) and \( y < 0 \)).
polynomial properties
Examining the properties of polynomial functions is crucial for understanding their graphs. For the function \( f(x) = 3x^2(x - 4)^4 \):

  • Degree: The total degree is 6 (2 from \( x^2 \)) + (4 from \( (x-4)^4 \)).
non-negative values
Understanding why \( f(x) \) does not take negative values is critical.

The function \( 3x^2(x - 4)^4 \) is non-negative because:
  • Square terms: \( x^2 \) and \( (x-4)^4 \) are both always non-negative for any real number \( x \).
  • Product property: Multiplying two non-negative terms results in a non-negative product.
This combination ensures that \( f(x) \) remains either positive or zero at all times. The non-negative nature is what mainly affects the location of the graph on the Cartesian plane.
In this case, understanding non-negative values helps explain why the graph does not touch or move into Quadrants III and IV, which require negative \( y \) values.

This ability to stay non-negative is a fundamental trait of polynomial functions involving even powers.

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