Question

name: tyler angoketto date: 1-12-26 per: 1a unit 5: polynomial functions homework 3: graphing polynomial functions this is a 2 - page document! directions: for each graph, (a) describe the end behavior, (b) determine whether its the graph of an even or odd degree function, and (c) determine the sign of the leading coefficient. 1. graph of a polynomial function a) b) odd c) positive 2. graph of a polynomial function a) b) even c) negative 3. using proper notation, describe the end behavior of a 14th degree polynomial with a positive leading coefficient. it would be 14 = even 4. using proper notation, describe the end behavior of a 9th degree polynomial with a negative leading coefficient. 9 = odd negative directions: graph each function and give its key characteristics. use a graphing calculator for the turning points and round to the nearest hundredth. 5. $f(x)=-x^{3}+6x^{2}-10x + 4$ graph of the function on a grid domain: $-infty infty$ range: $-infty infty$ rel. maximum(s): $1, - 1$ rel. minimum(s): $(3, - 2)$ end behavior: as $x
ightarrow-infty$, $f(x)
ightarrowinfty$ as $x
ightarrowinfty$, $f(x)
ightarrow-infty$ inc. intervals: $(1, infty)$ $(3, infty)$ dec. intervals: $(1, 3)$

Explanation:

Step1: Verify Q1 end behavior

As $x \to -\infty$, $f(x) \to -\infty$; as $x \to +\infty$, $f(x) \to +\infty$

Step2: Verify Q1 degree parity

End behavior is opposite, so odd degree

Step3: Verify Q1 leading coefficient

Right end rises, so positive leading coefficient

Step4: Verify Q2 end behavior

As $x \to -\infty$, $f(x) \to -\infty$; as $x \to +\infty$, $f(x) \to -\infty$

Step5: Verify Q2 degree parity

End behavior matches, so even degree

Step6: Verify Q2 leading coefficient

Both ends fall, so negative leading coefficient

Step7: Q3 end behavior (14th deg, + lead)

14 is even, + lead: as $x \to \pm\infty$, $f(x) \to +\infty$

Step8: Q4 end behavior (9th deg, - lead)

9 is odd, - lead: as $x \to -\infty$, $f(x) \to +\infty$; as $x \to +\infty$, $f(x) \to -\infty$

Step9: Q5 key characteristics

Step9a: Domain of polynomial

All real numbers: $(-\infty, \infty)$

Step9b: Range of odd-degree poly

All real numbers: $(-\infty, \infty)$

Step9c: Calculate rel max/min

$f(x)=-x^3+6x^2-10x+4$
Derivative: $f'(x)=-3x^2+12x-10$
Solve $-3x^2+12x-10=0$:
$x=\frac{-12\pm\sqrt{144-120}}{-6}=\frac{-12\pm\sqrt{24}}{-6}=2\pm\frac{\sqrt{6}}{3}$
$x\approx0.85, 3.15$
$f(0.85)\approx-0.88$, $f(3.15)\approx4.06$

Step9d: Confirm end behavior

Odd degree, - lead: as $x \to -\infty$, $f(x) \to +\infty$; as $x \to +\infty$, $f(x) \to -\infty$

Step9e: Inc/Dec intervals

Inc where $f'(x)>0$: $(0.85, 3.15)$
Dec where $f'(x)<0$: $(-\infty, 0.85)\cup(3.15, \infty)$

Answer:

1.
a) As $x \to -\infty$, $f(x) \to -\infty$; as $x \to +\infty$, $f(x) \to +\infty$
b) Odd degree
c) Positive
2.
a) As $x \to -\infty$, $f(x) \to -\infty$; as $x \to +\infty$, $f(x) \to -\infty$
b) Even degree
c) Negative

1. As $x \to -\infty$, $f(x) \to +\infty$; as $x \to +\infty$, $f(x) \to +\infty$
2. As $x \to -\infty$, $f(x) \to +\infty$; as $x \to +\infty$, $f(x) \to -\infty$

5.
Domain: $(-\infty, \infty)$
Range: $(-\infty, \infty)$
Rel. Maximum(s): $(3.15, 4.06)$
Rel. Minimum(s): $(0.85, -0.88)$
End Behavior: As $x \to -\infty$, $f(x) \to +\infty$; As $x \to +\infty$, $f(x) \to -\infty$
Inc. Intervals: $(0.85, 3.15)$
Dec. Intervals: $(-\infty, 0.85) \cup (3.15, \infty)$