Power Functions Practice

1. Consider the polynomial function $q(x)=3x^4-5x^3-2x^2+x-18$.

   What is the end behavior of the graph of $q\ $?
   

   Choose 1 answer:

   1. $x \rightarrow \infty$, $q(x) \rightarrow \infty$, and as $x \rightarrow -\infty$, $q(x) \rightarrow \infty$.
   2. As $x \rightarrow \infty$, $q(x) \rightarrow -\infty$, and as $x \rightarrow -\infty$, $q(x) \rightarrow \infty$.
   3. As $x \rightarrow \infty$, $q(x) \rightarrow -\infty$, and as $x \rightarrow -\infty$, $q(x) \rightarrow -\infty$.

   4. As $x \rightarrow \infty$, $q(x) \rightarrow \infty$, and as $x \rightarrow -\infty$, $q(x) \rightarrow -\infty$.

2. Consider the polynomial function $g(x)=-x^4+2x^3+5x^2-1$.
   

   What is the end behavior of the graph of $g\ $?

   Choose 1 answer:

   1. As $x \rightarrow \infty$, $g(x) \rightarrow \infty$, and as $x \rightarrow -\infty$, $g(x) \rightarrow \infty$.
   2. As $x \rightarrow \infty$, $g(x) \rightarrow -\infty$, and as $x \rightarrow -\infty$, $g(x) \rightarrow \infty$.
   3. As $x \rightarrow \infty$, $g(x) \rightarrow -\infty$, and as $x \rightarrow -\infty$, $g(x) \rightarrow -\infty$.

   4. As $x \rightarrow \infty$, $g(x) \rightarrow \infty$, and as $x \rightarrow -\infty$, $g(x) \rightarrow -\infty$.

3. Consider the polynomial function $h(x)=3x^7+x^4+5x^3-80$.
   

   What is the end behavior of the graph of $h\ $?
   

   Choose 1 answer:

   1. As $x \rightarrow \infty$, $h(x) \rightarrow \infty$, and as $x \rightarrow -\infty$, $h(x) \rightarrow \infty$.
   2. As $x \rightarrow \infty$, $h(x) \rightarrow -\infty$, and as $x \rightarrow -\infty$, $h(x) \rightarrow \infty$.
   3. As $x \rightarrow \infty$, $h(x) \rightarrow -\infty$, and as $x \rightarrow -\infty$, $h(x) \rightarrow -\infty$.

   4. As $x \rightarrow \infty$, $h(x) \rightarrow \infty$, and as $x \rightarrow -\infty$, $h(x) \rightarrow -\infty$.

4. Consider the polynomial function $q(x)=-2x^8+5x^6-3x^5+50$.

   What is the end behavior of the graph of $q\ $?

   Choose 1 answer:

   1. As $x \rightarrow \infty$, $q(x) \rightarrow \infty$, and as $x \rightarrow -\infty$, $q(x) \rightarrow \infty$.
   2. As $x \rightarrow \infty$, $q(x) \rightarrow -\infty$, and as $x \rightarrow -\infty$, $q(x) \rightarrow \infty$.
   3. As $x \rightarrow \infty$, $q(x) \rightarrow -\infty$, and as $x \rightarrow -\infty$, $q(x) \rightarrow -\infty$.
   4. As $x \rightarrow \infty$, $q(x) \rightarrow \infty$, and as $x \rightarrow -\infty$, $q(x) \rightarrow -\infty$.

Source: Khan Academy, https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-graphs/x2ec2f6f830c9fb89:poly-end-behavior/e/determine-the-end-behavior-of-polynomials
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

1. Thinking about end behavior

   A polynomial can be seen as the sum of several monomial terms of different degrees. As the input variable gets positively or negatively larger, the lower-degree terms become insignificant in comparison to the term with the highest degree, which is called the leading term.

   Therefore, in order to determine the end behavior of a polynomial, we consider its leading term. In particular, whether its degree is odd or even, and whether its sign is positive or negative.

   The degree of the leading term

   In the case of $q$, the leading term is $3x^4$.
   

   Since the degree of the leading term, $4$, is even, the leading term behaves the same as $x$ gets both positively and negatively large. In both cases, $x^4$ goes to positive infinity.

   The sign of the leading term

   Since the coefficient of the leading term, $4$, is positive, the leading term, $3x^4$, goes to positive infinity in both directions.

   Simply put, as $x \rightarrow \infty$, $q(x) \rightarrow \infty$, and as $x \rightarrow -\infty$, $q(x) \rightarrow \infty$.

   The following is the end behavior of the graph of $q$ :
   

   As $x \rightarrow \infty$, $q(x) \rightarrow \infty$, and as $x \rightarrow -\infty$, $q(x) \rightarrow \infty$.

   ---

2. Thinking about end behavior

   A polynomial can be seen as the sum of several monomial terms of different degrees. As the input variable gets positively or negatively larger, the lower-degree terms become insignificant in comparison to the term with the highest degree, which is called the leading term.

   Therefore, in order to determine the end behavior of a polynomial, we consider its leading term. In particular, whether its degree is odd or even, and whether its sign is positive or negative.

   The degree of the leading term

   In the case of $g$, the leading term is $-x^4$.
   

   Since the degree of the leading term, $4$, is even, the leading term behaves the same as $x$ gets both positively and negatively large. In both cases, $x^4$ goes to positive infinity.

   The sign of the leading term

   However, since the coefficient of the leading term, $-1$, is negative, the leading term, $-x^4$, goes to negative infinity in both directions.

   Simply put, as $x \rightarrow \infty$, $g(x) \rightarrow -\infty$, and as $x \rightarrow -\infty$, $g(x) \rightarrow -\infty$.

   The following is the end behavior of the graph of $g$ :
   

   As $x \rightarrow \infty$, $g(x) \rightarrow -\infty$, and as $x \rightarrow -\infty$, $g(x) \rightarrow -\infty$.

   ---

3. Thinking about end behavior

   A polynomial can be seen as the sum of several monomial terms of different degrees. As the input variable gets positively or negatively larger, the lower-degree terms become insignificant in comparison to the term with the highest degree, which is called the leading term.

   Therefore, in order to determine the end behavior of a polynomial, we consider its leading term. In particular, whether its degree is odd or even, and whether its sign is positive or negative.

   The degree of the leading term

   In the case of $h$, the leading term is $3x^7$.

   Since the degree of the leading term, $7$, is odd, $x^7$ goes to positive infinity as $x$ gets positively large, and goes to negative infinity as $x$ gets negatively large.

   The sign of the leading term

   Since the coefficient of the leading term, $3$, is positive, the leading term, $3x^7$, goes to positive infinity as $x$ gets positively large, and goes to negative infinity as $x$ gets negatively large.
   

   Simply put, as $x \rightarrow \infty$, $h(x) \rightarrow \infty$, and as $x \rightarrow -\infty$, $h(x) \rightarrow -\infty$.

   The following is the end behavior of the graph of $h$ :

   As $x \rightarrow \infty$, $h(x) \rightarrow \infty$, and as $x \rightarrow -\infty$, $h(x) \rightarrow -\infty$.

   ---

4. Thinking about end behavior

   A polynomial can be seen as the sum of several monomial terms of different degrees. As the input variable gets positively or negatively larger, the lower-degree terms become insignificant in comparison to the term with the highest degree, which is called the leading term.

   Therefore, in order to determine the end behavior of a polynomial, we consider its leading term. In particular, whether its degree is odd or even, and whether its sign is positive or negative.

   The degree of the leading term

   In the case of $q$, the leading term is $-2x^8$.

   Since the degree of the leading term, $8$, is even, the leading term behaves the same as $x$ gets both positively and negatively large. In both cases, $x^8$ goes to positive infinity.

   The sign of the leading term

   However, since the coefficient of the leading term, $-2$, is negative, the leading term, $-2x^8$, goes to negative infinity in both directions.

   Simply put, as $x \rightarrow \infty$, $q(x) \rightarrow -\infty$, and as $x \rightarrow -\infty$, $q(x) \rightarrow -\infty$.

   The following is the end behavior of the graph of $q$ :
   

   As $x \rightarrow \infty$, $q(x) \rightarrow -\infty$, and as $x \rightarrow -\infty$, $q(x) \rightarrow -\infty$.