Solution Bank Polynomials Target B
Answers to the Practice Problems are below in random order.
Answers to the Practice Problems are below in random order.
- The x part of the end behavior and the Domain will look the same for polynomial functions as the function has a Domain of all Reals. The y part of the end behavior and the Range will depend on whether the function is even or odd.
- Sometimes; this is true of an even degree polynomial with a positive leading coefficient
- Sometimes; although it will always have 3 solutions, one of the solutions may be a multiple root or 2 of the solutions may be complex solutions and therefore not an x-intercept
- Never; since a quartic is an even degree, the ends of the graph must be in the same direction and the range will either be \((-\infty,max] or [min,\infty)\).
- As \(\:x\ \longrightarrow\ -\infty,\ f\left(x\right)\ \longrightarrow\ +\infty,\) as \(\: x\ \longrightarrow\ +\infty,\ f\left(x\right)\ \longrightarrow\ -\infty\)
- As \(\:x\ \longrightarrow\ -\infty,\ f\left(x\right)\ \longrightarrow\ -\infty,\) as \(\: x\ \longrightarrow\ +\infty,\ f\left(x\right)\ \longrightarrow\ +\infty\)
- As \(\:x\ \longrightarrow\ -\infty,\ f\left(x\right)\ \longrightarrow\ +\infty,\) as \(\: x\ \longrightarrow\ +\infty,\ f\left(x\right)\ \longrightarrow\ +\infty\)
\(\begin{align}&A\ & \ \ & B\ & \ \ & C\\\\
&\text{Degree 3; LC is negative}\ & \ \ &\text{Degree 4; LC is negative}\ & \ \ &\text{Degree 5; LC is positive}\ & \ \ &\text{Degree 5; LC is negative}\end{align}\)
Degree: \(4\)
End Behavior: As \(\:x\ \longrightarrow\ -\infty,\ f\left(x\right)\ \longrightarrow\ +\infty,\) as \(\: x\ \longrightarrow\ +\infty,\ f\left(x\right)\ \longrightarrow\ +\infty\)
X-intercepts: \((-0.639,0)\) and \((0.435,0)\) both have multiplicity of \(1\)
Y-intercept: \((0,-4)\)
Maxima: \(5.022\) (local)
Minima: \(2.207\) (local); \(-4\) (absolute)
Domain: \((-\infty,\infty)\) Range: \([-4,\infty)\)
Degree: \(4\)
LC: \(-\frac{1}{2}\)
As \(\:x\ \longrightarrow\ -\infty,\ g\left(x\right)\ \longrightarrow\ -\infty,\) as \(\: x\ \longrightarrow\ +\infty,\ g\left(x\right)\ \longrightarrow\ -\infty\)
X-intercepts: \((-2,0)\ (1,0)\ (3,0)\)
Y-intercepts: \((0,9)\)
Degree: \(6\)
LC: \(1\)
As \(\:x\ \longrightarrow\ -\infty,\ g\left(x\right)\ \longrightarrow\ +\infty,\) as \(\: x\ \longrightarrow\ +\infty,\ g\left(x\right)\ \longrightarrow\ +\infty\)
X-intercepts: \((-1,0)\ (2,0)\ (4,0)\)
Y-intercepts: \((0,64)\)