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For the polynomial $y=2(x+5)(x-1)^2$, 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)
  • $\lim _{x \rightarrow-\infty} f(x)=-\infty$ 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)
  • $\lim _{x \rightarrow-\infty} f(x)=-\infty$ (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)



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


BList all $x$-intercepts, or state if there are no $x$-intercepts.


CList all $x$-intercepts where the graph crosses the $x$-axis, or state if there are none.


DList all $x$-intercepts where the graph touches the $x$-axis and turns, or state if there are none.


EThe $y$-intercept (give the $y$-value only).

Verified Answer :

  • The polynomial \(f(x) = 2(x+5)(x-1)^2\) has a leading term of \(2x^3\), meaning the graph falls to \(-\infty\) as \(x \rightarrow -\infty\) and rises to \(\infty\) as \(x \rightarrow \infty\).
  • The \(x\)-intercepts are at \(x = -5\) and \(x = 1\).
  • The graph crosses the \(x\)-axis at \(x = -5\) and touches/turns at \(x = 1\).
  • The \(y\)-intercept is \(10\).
  • Final answers: a. \(\lim _{x \rightarrow-\infty} f(x)=-\infty\) and \(\lim _{x \rightarrow \infty} f(x)=\infty\), b. \(\boxed{-5, 1}\), c. \(\boxed{-5}\), d. \(\boxed{1}\), e. \(\boxed{10}\).

Explanation

  1. End Behavior Analysis

We are given the polynomial function \(f(x) = 2(x+5)(x-1)^2\). To understand the graph of this function, we first determine its degree and leading behavior. Expanding the polynomial, the leading term is determined by the highest powers of \(x\), which is \(2 \cdot x \cdot x^2 = 2x^3\). Since the degree is \(n=3\) (an odd degree) and the leading coefficient is \(2\) (a positive value), the end behavior is determined by the limits as \(x\) approaches infinity or negative infinity. As \(x \rightarrow -\infty\), \(x^3 \rightarrow -\infty\), so \(f(x) \rightarrow -\infty\). As \(x \rightarrow \infty\), \(x^3 \rightarrow \infty\), so \(f(x) \rightarrow \infty\). Thus, the function falls toward negative infinity on the left and rises toward infinity on the right.

  1. Determining x-intercepts

The \(x\)-intercepts are the points where the function \(f(x) = 0\). Setting the polynomial to zero gives \(2(x+5)(x-1)^2 = 0\). This equation is satisfied when \(x+5 = 0\) or \((x-1)^2 = 0\). Solving these equations, we find the roots are \(x = -5\) and \(x = 1\). Therefore, the list of \(x\)-intercepts is \(-5, 1\).

  1. Behavior at x-intercepts

The behavior at each \(x\)-intercept depends on the multiplicity of the root. A root with an odd multiplicity causes the graph to cross the \(x\)-axis, while a root with an even multiplicity causes the graph to touch the \(x\)-axis and turn.

  • The root \(x = -5\) has a multiplicity of 1 (an odd number), so the graph crosses the \(x\)-axis at \(x = -5\).
  • The root \(x = 1\) has a multiplicity of 2 (an even number), so the graph touches the \(x\)-axis and turns at \(x = 1\).
  1. Finding the y-intercept

To find the \(y\)-intercept, we evaluate the function at \(x = 0\):
\[f(0) = 2(0+5)(0-1)^2 = 2(5)(-1)^2 = 2 \times 5 \times 1 = 10.\]
Thus, the \(y\)-intercept is \(10\).

Examples

Understanding the properties of polynomials, such as intercepts and end behavior, is crucial in fields like physics and engineering to model physical processes. For instance, if you are modeling the trajectory of a projectile or the energy output of a machine over time, identifying the roots tells you when the output is zero (e.g., when a projectile hits the ground), and analyzing the end behavior helps predict long-term performance. These graphical characteristics allow scientists to quickly interpret data patterns and make informed decisions about system designs.

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