QuestionAnalyze the function $f(x)=(x+3)(x-4)(x+6)$ : (a) Determine end behavior; (b) Find $x$ - and $y$ -intercepts.

Studdy Solution

STEP 1

Assumptions1. The polynomial function is $f(x)=(x+3)(x-4)(x+6)$

STEP 2

To determine the end behavior of the graph of the function, we need to look at the highest degree of the polynomial. In this case, it's a cubic polynomial because when we multiply out the factors, the highest power of $x$ will be.

STEP 3

For a cubic polynomial, the end behavior of the function is determined by the sign of the coefficient of the highest degree term. If the coefficient is positive, the function will rise to the right and fall to the left. If the coefficient is negative, the function will fall to the right and rise to the left.

STEP 4

In our case, if we multiply out the factors, the coefficient of the $x^{3}$ term will be positive (since we have an odd number of negative signs). Therefore, the graph of the function behaves like $y=x^{3}$ for large values of $|x|$ . This means the function will rise to the right and fall to the left.

STEP 5

To find the x-intercepts of the graph of the function, we need to set the function equal to zero and solve for $x$ .
$0 = (x+3)(x-4)(x+)$

STEP 6

The x-intercepts are the solutions to the equation. We can find them by setting each factor equal to zero and solving for $x$ .
$x+3 =0$ $x-4 =0$ $x+6 =0$

STEP 7

olving these equations gives us the x-intercepts.
$x = -3,4, -6$

STEP 8

To find the y-intercept of the graph of the function, we need to set $x=0$ and solve for $y$ .
$y = (0+3)(0-4)(0+6)$

STEP 9

olving this equation gives us the y-intercept.
$y =3 \times -4 \times6 = -72$ So, the end behavior of the graph of the function is that it behaves like $y=x^{3}$ for large values of $|x|$ , the x-intercepts are $x = -3,4, -6$ , and the y-intercept is $y = -72$ .

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QuestionAnalyze the function $f(x)=(x+3)(x-4)(x+6)$ : (a) Determine end behavior; (b) Find $x$ - and $y$ -intercepts.

Studdy Solution

STEP 1

Assumptions1. The polynomial function is $f(x)=(x+3)(x-4)(x+6)$

STEP 2

To determine the end behavior of the graph of the function, we need to look at the highest degree of the polynomial. In this case, it's a cubic polynomial because when we multiply out the factors, the highest power of $x$ will be.

STEP 3

STEP 4

STEP 5

To find the x-intercepts of the graph of the function, we need to set the function equal to zero and solve for $x$ .
$0 = (x+3)(x-4)(x+)$

STEP 6

The x-intercepts are the solutions to the equation. We can find them by setting each factor equal to zero and solving for $x$ .
$x+3 =0$ $x-4 =0$ $x+6 =0$

STEP 7

olving these equations gives us the x-intercepts.
$x = -3,4, -6$

STEP 8

To find the y-intercept of the graph of the function, we need to set $x=0$ and solve for $y$ .
$y = (0+3)(0-4)(0+6)$

STEP 9

olving this equation gives us the y-intercept.
$y =3 \times -4 \times6 = -72$ So, the end behavior of the graph of the function is that it behaves like $y=x^{3}$ for large values of $|x|$ , the x-intercepts are $x = -3,4, -6$ , and the y-intercept is $y = -72$ .

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QuestionAnalyze the function f(x)=(x+3)(x−4)(x+6)f(x)=(x+3)(x-4)(x+6)f(x)=(x+3)(x−4)(x+6): (a) Determine end behavior; (b) Find xxx- and yyy-intercepts.

Studdy Solution

STEP 1

Assumptions1. The polynomial function is f(x)=(x+3)(x−4)(x+6)f(x)=(x+3)(x-4)(x+6)f(x)=(x+3)(x−4)(x+6)

STEP 2

To determine the end behavior of the graph of the function, we need to look at the highest degree of the polynomial. In this case, it's a cubic polynomial because when we multiply out the factors, the highest power of xxx will be.

STEP 3

STEP 4

STEP 5

To find the x-intercepts of the graph of the function, we need to set the function equal to zero and solve for xxx.0=(x+3)(x−4)(x+)0 = (x+3)(x-4)(x+)0=(x+3)(x−4)(x+)

STEP 6

The x-intercepts are the solutions to the equation. We can find them by setting each factor equal to zero and solving for xxx.x+3=0x+3 =0x+3=0x−4=0x-4 =0x−4=0x+6=0x+6 =0x+6=0

STEP 7

olving these equations gives us the x-intercepts.x=−3,4,−6x = -3,4, -6x=−3,4,−6

STEP 8

To find the y-intercept of the graph of the function, we need to set x=0x=0x=0 and solve for yyy.y=(0+3)(0−4)(0+6)y = (0+3)(0-4)(0+6)y=(0+3)(0−4)(0+6)

STEP 9

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QuestionAnalyze the function $f(x)=(x+3)(x-4)(x+6)$ : (a) Determine end behavior; (b) Find $x$ - and $y$ -intercepts.

Assumptions1. The polynomial function is $f(x)=(x+3)(x-4)(x+6)$

To determine the end behavior of the graph of the function, we need to look at the highest degree of the polynomial. In this case, it's a cubic polynomial because when we multiply out the factors, the highest power of $x$ will be.

To find the x-intercepts of the graph of the function, we need to set the function equal to zero and solve for $x$ .
$0 = (x+3)(x-4)(x+)$

The x-intercepts are the solutions to the equation. We can find them by setting each factor equal to zero and solving for $x$ .
$x+3 =0$ $x-4 =0$ $x+6 =0$

olving these equations gives us the x-intercepts.
$x = -3,4, -6$

To find the y-intercept of the graph of the function, we need to set $x=0$ and solve for $y$ .
$y = (0+3)(0-4)(0+6)$