1) Sketch one degree \(4\) polynomial that is an even function and one degree \(4\) polynomial that is not an even function.
2) Explain why an odd degree polynomial can never be an even function. Address characteristics of the graphs such as symmetry and end behavior in your explanation.
3) Given \(g(x)=\frac{1}{4}x^4+ax^2+6\), explain why for all real values of \(a\), \(g(x)\) must be an even function.
For problems 4-7, determine if the graph is even, odd, or neither.
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2) Explain why an odd degree polynomial can never be an even function. Address characteristics of the graphs such as symmetry and end behavior in your explanation.
3) Given \(g(x)=\frac{1}{4}x^4+ax^2+6\), explain why for all real values of \(a\), \(g(x)\) must be an even function.
For problems 4-7, determine if the graph is even, odd, or neither.
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For problems 8-13, algebraically determine if the functions are even, odd, or neither.
8) \(f(x)=-2x^4+5x^3-1\)
9) \(g(x)=7x^5-x^3+6\)
10) \(h(x)=3x^6-9x^2-8\)
11) \(j(x)=-4x^5+2x^3-7x\)
12) \(k(x)=-x^4+2x^2-4\)
13) \(m(x)=6x^5-x^3+3x\)
14) Given the function \(f(x)=3x^4-6x^2-1\), complete the following:
a) Is \(f(x)\) an even function, and odd function, or neither?
b) Determine the degree and leading coefficient.
c) Determine the end behavior.
d) Use a graphing device to graph \(n(x)\). Use the graph to determine the domain and range.
15) Given the function \(f(x)=-x^5-6x^3-2x\), complete the following:
a) Is \(f(x)\) an even function, and odd function, or neither?
b) Determine the degree and leading coefficient.
c) Determine the end behavior.
d) Without graphing, determine the domain and range.
Review
16) Rewrite as an equivalent expression in simplest form: \(\large{\frac{(-3x^5y^{-3})^4}{(3x^{-4}y^{-6})^2}}\).
17) Factor the following expressions completely over the real numbers:
a) \(9x^4-64\)
b) \(x^4-9x^2+20\)
c) \(8x^3-27\)
d) \(3x^3+4x^2-27x-36\)
18) Given that \(x=6\) is a root of \(f(x)=x^3-2x^2-15x-54\), find the remaining roots.
Solution Bank