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For the polynomial $y=-2(x-4)^4(x+10)$, find the following.


AThe end behavior of the graph:
  • $\lim _{x \rightarrow-\infty} f(x)=\infty$ and $\lim _{x \rightarrow \infty} f(x)=\infty$ (the function rises toward infinity on both the left and right).
  • $\lim _{x \rightarrow-\infty} f(x)=-\infty$ (the function falls toward negative infinity on the left), and $\lim _{x \rightarrow \infty} f(x)=\infty$ (the function rises toward infinity on the right).
  • $\lim _{x \rightarrow-\infty} f(x)=-\infty$ and $\lim _{x \rightarrow \infty} f(x)=-\infty$ (the function falls toward negative infinity on both the left and right).
  • $\lim _{x \rightarrow-\infty} f(x)=\infty$ (the function rises toward infinity on the left), and $\lim _{x \rightarrow \infty} f(x)=-\infty$ (the function falls toward negative infinity on the right).



When listing $x$-intercepts, give $x$-values only, separated by commas. Round decimals to 1 decimal place.


B
  • List all $x$-intercepts: [ ]
  • No $x$-intercepts



c.

  • List all $x$-intercepts where the graph crosses the $x$-axis: [ ]
  • No $x$-intercepts where the graph crosses the $x$-axis



d.

  • List all $x$-intercepts where the graph touches the $x$-axis and turns: [ ]
  • No $x$-intercepts where the graph touches the $x$-axis and turns


EThe $y$-intercept ($y$-value only): [ ]

Verified Answer :

  • The end behavior is \(\lim _{x \rightarrow -\infty} f(x) = \infty\) and \(\lim _{x \rightarrow \infty} f(x) = -\infty\) because the degree is odd (5) and the leading coefficient is negative (-2).
  • The x-intercepts are \(x = -10\) (where the graph crosses the x-axis) and \(x = 4\) (where the graph touches and turns).
  • The y-intercept is \(f(0) = -5120\).
  • Final summary: a. Last option; b. -10, 4; c. -10; d. 4; e. -5120.

Explanation

  1. Analyzing End Behavior

The given polynomial is \(f(x) = -2(x - 4)^4(x + 10)\). To determine the end behavior, we look at the leading coefficient and the degree of the polynomial. The degree is \(n = 4 + 1 = 5\), which is odd. The leading coefficient is \(a_n = -2\), which is negative. For a polynomial of odd degree with a negative leading coefficient, as \(x \to \infty\), \(f(x) \to -\infty\), and as \(x \to -\infty\), \(f(x) \to \infty\). Therefore, the end behavior is: \(\lim _{x \rightarrow -\infty} f(x) = \infty\) and \(\lim _{x \rightarrow \infty} f(x) = -\infty\).

  1. Identifying x-intercepts

To find the x-intercepts, we set \(f(x) = 0\):
\[-2(x - 4)^4(x + 10) = 0.\]
This occurs when \(x - 4 = 0\) or \(x + 10 = 0\), which gives the x-intercepts at \(x = 4\) and \(x = -10\).

  1. Determining Graph Behavior at x-intercepts

The behavior of the graph at an x-intercept depends on the multiplicity of the root. The root \(x = -10\) has a multiplicity of 1, which is odd, meaning the graph crosses the x-axis at \(x = -10\). The root \(x = 4\) has a multiplicity of 4, which is even, meaning the graph touches the x-axis and turns at \(x = 4\).

  1. Calculating the y-intercept

The y-intercept occurs at \(f(0)\). We calculate this as:
\[f(0) = -2(0 - 4)^4(0 + 10) = -2(256)(10) = -5120.\]
Thus, the y-intercept is \(-5120\).

Examples

The study of polynomial end behavior and intercepts is essential in fields like physics and engineering for modeling real-world phenomena. For instance, in structural engineering, the shape of a load-bearing beam can often be represented by polynomial functions. Knowing where the curve crosses or touches specific axes helps engineers identify critical points of inflection, maximum displacement, or zero-stress zones, ensuring the safety and stability of the designed structure. Understanding these mathematical properties allows for precise predictions and efficient design solutions in practical engineering tasks.

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