A) Graph y1=x^3 y2=x^3-3 x^2-3 b) In both graphs,...

a) Graph $$ \begin{aligned} &y_{1}=x^{3} \\ &y_{2}=x^{3}-3 x^{2}-3 \end{aligned} $$ b) In both graphs, for values of $x>3$, do the functions increase or decrease as $x$ increases? c) When the leading term of a polynomial function is $x^{3}$, the polynomial must increase as $x$ increases for $x>a$, where $a$ is some real number greater than 0 . Explain why this must be so. d) In both graphs, for values of $x<-3$, do the functions increase or decrease as $x$ decreases? e) When the leading term of a polynomial function is $x^{3}$, the polynomial must decrease as $x$ decreases for $x<a$, where $a$ is some real number less than 0 . Explain why this must be so.

a) Graph
$$
\begin{aligned}
&y_{1}=x^{3} \\
&y_{2}=x^{3}-3 x^{2}-3
\end{aligned}
$$
b) In both graphs, for values of $x>3$, do the functions increase or decrease as $x$ increases?
c) When the leading term of a polynomial function is $x^{3}$, the polynomial must increase as $x$ increases for $x>a$, where $a$ is some real number greater than 0 . Explain why this must be so.
d) In both graphs, for values of $x<-3$, do the functions increase or decrease as $x$ decreases?
e) When the leading term of a polynomial function is $x^{3}$, the polynomial must decrease as $x$ decreases for $x<a$, where $a$ is some real number less than 0 . Explain why this must be so.

Key Concepts

Dominance of the Leading Term

In any polynomial function, the term with the highest degree is crucial in determining the behavior of the function for large absolute values of x. This term overshadows the contribution of lower degree terms as x becomes very large or very small, ensuring that the sign and degree of the leading term control the overall increasing or decreasing trend of the function at the extremes.

End Behavior of Polynomial Functions

The end behavior of a polynomial function describes how the function behaves as x approaches large positive or negative values. In polynomials with a leading term of x³, for sufficiently large positive x the cubed term dominates and causes the function to increase, while for sufficiently large negative x, it causes the function to decrease. This dominance of the highest power term means that any additional lower degree terms have a negligible effect on the behavior at the extremes of the domain.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to nonnegative integer powers, combined using addition, subtraction, and multiplication. Their graphs can exhibit various behaviors—such as increasing, decreasing, and having turning points—which are largely influenced by the degree of the polynomial and the coefficients of its terms.

Leading Coefficient Test

The leading coefficient test is a method to predict the end behavior of a polynomial function by looking at its highest power term. For a polynomial with an odd degree and a positive leading coefficient, the function increases without bound as x becomes large (x ? +?) and decreases without bound as x becomes very negative (x ? -?). This explains why, for functions with a leading term of x³, the behavior is consistent regardless of other lower degree terms.

Transcript

Please give Ace some feedback