Question

1. reasoning the end behavior of a polynomial function f is described by f(x) → +∞ as x → −∞ and f(x) → −∞ as x → +∞. describe the end behavior of g(x) = −f(x). justify your answer.

1. patterns use technology to graph f(x) = x², g(x) = x⁴, and h(x) = x⁶ in the same coordinate plane.

a. what do you notice about the functions and their points of intersection? does the pattern continue for greater, even powers of x? explain.

b. is there a similar pattern for functions with odd powers of x? explain.

Explanation:

Problem 43

Step1: Analyze transformation

The function \( g(x) = -f(x) \) is a reflection of \( f(x) \) over the x - axis. For any function \( y = f(x) \), the transformation \( y=-f(x) \) changes the sign of the function's output.

Step2: Find end behavior of \( g(x) \)

We know that as \( x
ightarrow-\infty \), \( f(x)
ightarrow+\infty \). When we apply the transformation \( g(x)=-f(x) \), we multiply the output of \( f(x) \) by - 1. So, as \( x
ightarrow-\infty \), \( g(x)=-f(x)
ightarrow - (+\infty)=-\infty \).
We also know that as \( x
ightarrow+\infty \), \( f(x)
ightarrow-\infty \). Then, as \( x
ightarrow+\infty \), \( g(x)=-f(x)
ightarrow - (-\infty)=+\infty \).

Step1: Graph the functions

Using a graphing utility (such as a graphing calculator or software like Desmos), we graph \( f(x)=x^{2} \), \( g(x)=x^{4} \), and \( h(x)=x^{6} \).

• The graph of \( f(x) = x^{2}\) is a parabola opening upwards with vertex at the origin.
• The graph of \( g(x)=x^{4}\) is also a curve opening upwards with vertex at the origin. For \(|x|>1\), \(x^{4}>x^{2}\) (since if \(x = 2\), \(x^{2}=4\) and \(x^{4}=16\); if \(x=- 2\), \(x^{2}=4\) and \(x^{4}=16\)). For \(|x| = 1\), \(x^{4}=x^{2}=1\). For \(0<|x|<1\), \(x^{4}
• The graph of \( h(x)=x^{6}\) is a curve opening upwards with vertex at the origin. For \(|x|>1\), \(x^{6}>x^{4}\) (if \(x = 2\), \(x^{4}=16\) and \(x^{6}=64\); if \(x=-2\), \(x^{4}=16\) and \(x^{6}=64\)). For \(|x| = 1\), \(x^{6}=x^{4}=1\). For \(0<|x|<1\), \(x^{6}

All three functions intersect at \(x=- 1\), \(x = 0\), and \(x = 1\) (since \( (-1)^{2}=(-1)^{4}=(-1)^{6}=1\), \(0^{2}=0^{4}=0^{6}=0\), and \(1^{2}=1^{4}=1^{6}=1\)).

Step2: Analyze the pattern for greater even powers

For a general even - powered function \(y = x^{n}\) where \(n = 2k\), \(k\in\mathbb{N}\) and \(n_1=2k_1\), \(n_2 = 2k_2\) with \(k_2>k_1\) (so \(n_2>n_1\)):

• When \(|x|>1\), we can use the property of exponents. If \(|x|>1\), then \(x^{n_2}=x^{2k_2}=(x^{2})^{k_2}\) and \(x^{n_1}=x^{2k_1}=(x^{2})^{k_1}\). Since \(x^{2}>1\) when \(|x|>1\), and for \(a > 1\) and \(m>n\), \(a^{m}>a^{n}\), we have \(x^{n_2}>x^{n_1}\) when \(|x|>1\) and \(n_2>n_1\) (both even).
• When \(|x| = 1\), \(x^{n}=1\) for any even \(n\), so all even - powered functions \(y = x^{n}\) intersect at \(x=\pm1\) and \(x = 0\) (since \(0^{n}=0\) for any non - negative integer \(n\)).
• When \(0<|x|<1\), if \(|x|<1\), then \(x^{2}<1\). Let \(a=x^{2}\), \(0 < a<1\) and \(m>n\) (positive integers). For \(0 < a<1\) and \(m>n\), \(a^{m}n_1\) (both even).

So the pattern is: All these even - powered functions (\(y=x^{2}\), \(y = x^{4}\), \(y=x^{6}\)) intersect at \(x=-1\), \(x = 0\), and \(x = 1\). For \(|x|>1\), the function with the larger even power has a greater value, and for \(0<|x|<1\), the function with the smaller even power has a greater value. This pattern continues for greater even powers of \(x\) (i.e., for \(y=x^{2k}\) where \(k\) is a positive integer and \(k\) increases) because of the properties of exponents for numbers greater than 1, between 0 and 1, and equal to 1.

Let's consider odd - powered functions, for example, \(y=x\), \(y = x^{3}\), \(y=x^{5}\).

• Graph the functions: The graph of \(y = x\) is a straight line passing through the origin with a slope of 1. The graph of \(y=x^{3}\) is a curve passing through the origin, increasing for all real \(x\), and symmetric about the origin. The graph of \(y = x^{5}\) is also a curve passing through the origin, increasing for all real \(x\), and symmetric about the origin.
• Analyze the intersection and behavior:
• Intersection: All odd - powered functions \(y=x^{n}\) (where \(n\) is odd) pass through the origin \((0,0)\) and intersect at \(x=-1\) (since \((-1)^{n}=-1\) for odd \(n\)) and \(x = 1\) (since \(1^{n}=1\) for any \(n\)).
• For \(x>1\): Let \(n_1\) and \(n_2\) be odd positive integers with \(n_2>n_1\). If \(x > 1\), then \(x^{n_2}>x^{n_1}\). For example, if \(x = 2\), \(n_1 = 1\), \(n_2=3\), \(2^{3}=8>2^{1}=2\); if \(n_1 = 3\), \(n_2 = 5\), \(2^{5}=32>2^{3}=8\).
• For \(0n_1\) (both odd), then \(x^{n_2}
• For \(x<-1\): Let \(x=-a\) where \(a > 1\). Then \(x^{n}=(-a)^{n}=-a^{n}\) (since \(n\) is odd). If \(n_2>n_1\) (both odd), then \(a^{n_2}>a^{n_1}\), so \(-a^{n_2}<-a^{n_1}\), or \(x^{n_2}n_1\) (both odd).
• For \(-1n_1\) (both odd), then \(a^{n_2}n_1\)), so \(-a^{n_2}>-a^{n_1}\), or \(x^{n_2}>x^{n_1}\) when \(-1n_1\) (both odd).

So there is a similar pattern for functions with odd powers of \(x\). The odd - powered functions intersect at \(x=-1\), \(x = 0\), and \(x = 1\). For \(x>1\) or \(-1

Answer:

As \( x
ightarrow-\infty \), \( g(x)
ightarrow-\infty \) and as \( x
ightarrow+\infty \), \( g(x)
ightarrow+\infty \). This is because \( g(x)=-f(x) \) reflects \( f(x) \) over the x - axis, which reverses the end - behavior of \( f(x) \).

Problem 45a