Math / Algebra

QuestionFor the polynomial $y=-2(x-5)^{4}(x+8)$ , find the following: a. The end behavior of the graph. Rises left and right Falls left and rises right Rises left and falts right Falls left and right When listing $x$ -intercepts, give $x$ -values only separated by commas. Round decimals to 1 decimal place. b. List all x-intercepts: $\square$ No x-intercepts c. List all $x$ -intercepts where graph crosses $x$ -axis: $\square$ No $x$ -intercepts where graph crosses $x$ -axis d. List all $x$ -intercepts where graph touches $x$ -axis and turns: $\square$ No $x$ -intercepts where graph touches $x$ -axis and turns e. The $y$ -intercept. ( $y$ -value only.) $\square$

Studdy Solution

STEP 1

1. The polynomial is given in factored form: $y = -2(x-5)^4(x+8)$ .
2. The degree of the polynomial and the leading coefficient determine the end behavior.
3. The $x$ -intercepts are found by setting $y = 0$ .
4. The behavior at each $x$ -intercept depends on the multiplicity of the factor.
5. The $y$ -intercept is found by setting $x = 0$ .

STEP 2

1. Determine the end behavior of the polynomial.
2. Find the $x$ -intercepts.
3. Classify the $x$ -intercepts as crossing or touching and turning.
4. Determine the $y$ -intercept.

STEP 3

The degree of the polynomial is $4 + 1 = 5$ , which is odd.
The leading term is $-2x^5$ , which has a negative coefficient.
For an odd-degree polynomial with a negative leading coefficient, the end behavior is: Falls left and rises right.

STEP 4

To find $x$ -intercepts, set $y = 0$ :
$-2(x-5)^4(x+8) = 0$
The solutions are $x = 5$ and $x = -8$ .

STEP 5

Examine the multiplicity of each factor:
- $(x-5)^4$ has even multiplicity (4), so the graph touches and turns at $x = 5$ . - $(x+8)$ has odd multiplicity (1), so the graph crosses the $x$ -axis at $x = -8$ .

STEP 6

To find the $y$ -intercept, set $x = 0$ :
$y = -2(0-5)^4(0+8) = -2(625)(8) = -10000$
The answers are:
a. Falls left and rises right
b. List all $x$ -intercepts: $5, -8$
c. List all $x$ -intercepts where graph crosses $x$ -axis: $-8$
d. List all $x$ -intercepts where graph touches $x$ -axis and turns: $5$
e. The $y$ -intercept: $-10000$

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Studdy Solution

STEP 1

STEP 2

1. Determine the end behavior of the polynomial.
2. Find the $x$ -intercepts.
3. Classify the $x$ -intercepts as crossing or touching and turning.
4. Determine the $y$ -intercept.

STEP 3

The degree of the polynomial is $4 + 1 = 5$ , which is odd.
The leading term is $-2x^5$ , which has a negative coefficient.
For an odd-degree polynomial with a negative leading coefficient, the end behavior is: Falls left and rises right.

STEP 4

To find $x$ -intercepts, set $y = 0$ :
$-2(x-5)^4(x+8) = 0$
The solutions are $x = 5$ and $x = -8$ .

STEP 5

Examine the multiplicity of each factor:
- $(x-5)^4$ has even multiplicity (4), so the graph touches and turns at $x = 5$ . - $(x+8)$ has odd multiplicity (1), so the graph crosses the $x$ -axis at $x = -8$ .

STEP 6

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Studdy Solution

STEP 1

STEP 2

1. Determine the end behavior of the polynomial.2. Find the x x x-intercepts.3. Classify the x x x-intercepts as crossing or touching and turning.4. Determine the y y y-intercept.

STEP 3

The degree of the polynomial is 4+1=5 4 + 1 = 5 4+1=5, which is odd.The leading term is −2x5-2x^5−2x5, which has a negative coefficient.For an odd-degree polynomial with a negative leading coefficient, the end behavior is: Falls left and rises right.

STEP 4

To find x x x-intercepts, set y=0 y = 0 y=0: −2(x−5)4(x+8)=0-2(x-5)^4(x+8) = 0−2(x−5)4(x+8)=0The solutions are x=5 x = 5 x=5 and x=−8 x = -8 x=−8.

STEP 5

Examine the multiplicity of each factor:- (x−5)4 (x-5)^4 (x−5)4 has even multiplicity (4), so the graph touches and turns at x=5 x = 5 x=5. - (x+8) (x+8) (x+8) has odd multiplicity (1), so the graph crosses the x x x-axis at x=−8 x = -8 x=−8.

STEP 6

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1. Determine the end behavior of the polynomial.
2. Find the $x$ -intercepts.
3. Classify the $x$ -intercepts as crossing or touching and turning.
4. Determine the $y$ -intercept.

The degree of the polynomial is $4 + 1 = 5$ , which is odd.
The leading term is $-2x^5$ , which has a negative coefficient.
For an odd-degree polynomial with a negative leading coefficient, the end behavior is: Falls left and rises right.

To find $x$ -intercepts, set $y = 0$ :
$-2(x-5)^4(x+8) = 0$
The solutions are $x = 5$ and $x = -8$ .

Examine the multiplicity of each factor:
- $(x-5)^4$ has even multiplicity (4), so the graph touches and turns at $x = 5$ . - $(x+8)$ has odd multiplicity (1), so the graph crosses the $x$ -axis at $x = -8$ .