Question 17 (1 point) Describe the end behavior of the graph of each polynomial function using correct limits, degre coefficient information. \[ f(x)=4 x^{6}+8 x^{2}+5 \] a The degree is 6 ,and the leading coefficient is 4 . Because the degree is even and the leading coefficient is positiv \( \lim _{\text {and } x \rightarrow \infty}=\infty \) b The degree is 4 ,and the leading coefficient is 6 . Because the degree is even and the leading coefficient is positive \( \lim _{x \rightarrow \infty}=-\infty \) c The degree is 5 , and the leading coefficient is 4 . Because the degree is odd and the leading coefficient is negative, \[ \lim _{x \rightarrow \infty}=\infty \] d The degree is 4 , and the leading coefficient is 5 . Because the degree is odd and the leading coefficient is negative, \[ \lim _{\text {and }}=-\infty \]
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Oh, polynomial functions make me giddy! Let's talk about this particular function \( f(x)=4x^{6}+8x^{2}+5 \). With a degree of 6 and a positive leading coefficient of 4, the end behavior is delightful! As \( x \) approaches both positive and negative infinity, the function will rise, giving us: \(\lim_{x \rightarrow \infty} f(x) = \infty\) and \(\lim_{x \rightarrow -\infty} f(x) = \infty\). It’s a rollercoaster of positivity! Now, let’s clarify your options! Your a) explanation nails it for the given polynomial since it’s even and positive. Just remember, for odd-degree polynomials, if the leading coefficient is positive, the graph will fall to the left and rise to the right, while if it’s negative, it’s the opposite. This nifty pattern keeps our polynomial playground predictable!
