A) Graph y1=x^4 y2=x^4-6 x^2 b) In each graph, for...

a) Graph $$ \begin{aligned} &y_{1}=x^{4} \\ &y_{2}=x^{4}-6 x^{2} \end{aligned} $$ b) In each graph, for values of $x>3$, are the functions increasing or decreasing as $x$ increases? c) When the leading term of a polynomial function is $x^{4}$, 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 each graph, for values of $x<-3$, are the functions increasing or decreasing as $x$ decreases? e) When the leading term of a polynomial function is $x^{4}$, the polynomial must increase as $x$ decreases for $x<a$, where $a$ is some real number less than 0 . Explain why this must be so.

a) Graph
$$
\begin{aligned}
&y_{1}=x^{4} \\
&y_{2}=x^{4}-6 x^{2}
\end{aligned}
$$
b) In each graph, for values of $x>3$, are the functions increasing or decreasing as $x$ increases?
c) When the leading term of a polynomial function is $x^{4}$, 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 each graph, for values of $x<-3$, are the functions increasing or decreasing as $x$ decreases?
e) When the leading term of a polynomial function is $x^{4}$, the polynomial must increase as $x$ decreases for $x<a$, where $a$ is some real number less than 0 . Explain why this must be so.

Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions that involve sums of powers of x with constant coefficients. They are continuous and smooth curves that can be analyzed by observing features such as roots, turning points, and inflection points. Understanding polynomial functions is essential in graphing, as specific characteristics like degree and leading coefficient determine the overall shape of the graph.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting the function by identifying key features like intercepts, local extrema, and end behavior. By analyzing the degree and leading coefficient, one can sketch an accurate curve that represents the behavior of the function across its domain. Graphs help in visualizing how shifts or transformations, such as subtracting a multiple of x², modify the shape without altering the fundamental end behavior dictated by the highest power term.

End Behavior of Polynomial Functions

The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity. This behavior is primarily determined by the leading term of the polynomial. For polynomials with a leading term of x?, which is of even degree and positive, the function increases towards infinity for both large positive and large negative values of x, ensuring that the ends of the graph rise.

Leading Coefficient Test

The leading coefficient test is a useful tool for predicting the long-term behavior of polynomial functions. This test considers the degree and sign of the leading coefficient to determine whether the function’s graph rises or falls as x moves towards positive or negative infinity. For example, a positive leading coefficient in an even-degree polynomial such as x? guarantees that the function will eventually increase on both ends.

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