QuestionAnalyze the function $f(x)=(x+1)(x-2)(x+5)$ .
(a) What is the end behavior? (b) Find the $x$ - and $y$ -intercepts.

Studdy Solution

STEP 1

Assumptions1. The polynomial function is $f(x)=(x+1)(x-)(x+5)$ . We need to analyze the function in terms of its end behavior, and its $x$ and $y$ intercepts.

STEP 2

To determine the end behavior of the graph of the function, we need to look at the degree and the leading coefficient of the polynomial.
The degree of the polynomial is the highest power of $x$ . In this case, since we have three factors of $x$ , the degree of the polynomial is.
The leading coefficient of the polynomial is the coefficient of the highest degree term. In this case, since all the coefficients of $x$ in the factors are1, the leading coefficient is1.

STEP 3

The end behavior of a polynomial function is determined by the sign of the leading coefficient and the degree of the polynomial.
If the degree is odd and the leading coefficient is positive, the end behavior of the function is that it decreases to negative infinity as $x$ approaches negative infinity and increases to positive infinity as $x$ approaches positive infinity.
So, the end behavior of the function $f(x)=(x+1)(x-2)(x+5)$ is the same as the function $y=x^{3}$ .

STEP 4

Next, we need to find the $x$ -intercepts of the graph of the function. The $x$ -intercepts are the values of $x$ for which $f(x)=0$ .
So, we set the function equal to zero and solve for $x$ .
$(x+1)(x-2)(x+)=0$

STEP 5

The equation $(x+1)(x-2)(x+5)=0$ is true if any of the factors is equal to zero. So, we set each factor equal to zero and solve for $x$ .
$x+1=0, x-2=0, x+5=0$

STEP 6

olving the equations $x+1=0$ , $x-2=0$ , and $x+5=0$ gives the solutions $x=-1$ , $x=2$ , and $x=-5$ .
So, the $x$ -intercepts of the graph of the function are $x=-1$ , $x=2$ , and $x=-5$ .

STEP 7

Finally, we need to find the $y$ -intercept of the graph of the function. The $y$ -intercept is the value of $f(x)$ when $x=0$ .
So, we substitute $x=0$ into the function and solve for $f(x)$ .
$f(0)=(0+1)(0-2)(0+5)$

STEP 8

Calculating $f(0)=(0+1)(0-2)(0+5)$ gives $f(0)=-10$ .
So, the $y$ -intercept of the graph of the function is $y=-10$ .

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QuestionAnalyze the function $f(x)=(x+1)(x-2)(x+5)$ .
(a) What is the end behavior? (b) Find the $x$ - and $y$ -intercepts.

Studdy Solution

STEP 1

Assumptions1. The polynomial function is $f(x)=(x+1)(x-)(x+5)$ . We need to analyze the function in terms of its end behavior, and its $x$ and $y$ intercepts.

STEP 2

STEP 3

STEP 4

Next, we need to find the $x$ -intercepts of the graph of the function. The $x$ -intercepts are the values of $x$ for which $f(x)=0$ .
So, we set the function equal to zero and solve for $x$ .
$(x+1)(x-2)(x+)=0$

STEP 5

The equation $(x+1)(x-2)(x+5)=0$ is true if any of the factors is equal to zero. So, we set each factor equal to zero and solve for $x$ .
$x+1=0, x-2=0, x+5=0$

STEP 6

olving the equations $x+1=0$ , $x-2=0$ , and $x+5=0$ gives the solutions $x=-1$ , $x=2$ , and $x=-5$ .
So, the $x$ -intercepts of the graph of the function are $x=-1$ , $x=2$ , and $x=-5$ .

STEP 7

Finally, we need to find the $y$ -intercept of the graph of the function. The $y$ -intercept is the value of $f(x)$ when $x=0$ .
So, we substitute $x=0$ into the function and solve for $f(x)$ .
$f(0)=(0+1)(0-2)(0+5)$

STEP 8

Calculating $f(0)=(0+1)(0-2)(0+5)$ gives $f(0)=-10$ .
So, the $y$ -intercept of the graph of the function is $y=-10$ .

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QuestionAnalyze the function f(x)=(x+1)(x−2)(x+5)f(x)=(x+1)(x-2)(x+5)f(x)=(x+1)(x−2)(x+5). (a) What is the end behavior? (b) Find the xxx- and yyy-intercepts.

Studdy Solution

STEP 1

Assumptions1. The polynomial function is f(x)=(x+1)(x−)(x+5)f(x)=(x+1)(x-)(x+5)f(x)=(x+1)(x−)(x+5). We need to analyze the function in terms of its end behavior, and its xxx and yyy intercepts.

STEP 2

STEP 3

STEP 4

Next, we need to find the xxx-intercepts of the graph of the function. The xxx-intercepts are the values of xxx for which f(x)=0f(x)=0f(x)=0.So, we set the function equal to zero and solve for xxx.(x+1)(x−2)(x+)=0(x+1)(x-2)(x+)=0(x+1)(x−2)(x+)=0

STEP 5

The equation (x+1)(x−2)(x+5)=0(x+1)(x-2)(x+5)=0(x+1)(x−2)(x+5)=0 is true if any of the factors is equal to zero. So, we set each factor equal to zero and solve for xxx.x+1=0,x−2=0,x+5=0x+1=0, x-2=0, x+5=0x+1=0,x−2=0,x+5=0

STEP 6

olving the equations x+1=0x+1=0x+1=0, x−2=0x-2=0x−2=0, and x+5=0x+5=0x+5=0 gives the solutions x=−1x=-1x=−1, x=2x=2x=2, and x=−5x=-5x=−5.So, the xxx-intercepts of the graph of the function are x=−1x=-1x=−1, x=2x=2x=2, and x=−5x=-5x=−5.

STEP 7

Finally, we need to find the yyy-intercept of the graph of the function. The yyy-intercept is the value of f(x)f(x)f(x) when x=0x=0x=0.So, we substitute x=0x=0x=0 into the function and solve for f(x)f(x)f(x).f(0)=(0+1)(0−2)(0+5)f(0)=(0+1)(0-2)(0+5)f(0)=(0+1)(0−2)(0+5)

STEP 8

Calculating f(0)=(0+1)(0−2)(0+5)f(0)=(0+1)(0-2)(0+5)f(0)=(0+1)(0−2)(0+5) gives f(0)=−10f(0)=-10f(0)=−10.So, the yyy-intercept of the graph of the function is y=−10y=-10y=−10.

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QuestionAnalyze the function $f(x)=(x+1)(x-2)(x+5)$ .
(a) What is the end behavior? (b) Find the $x$ - and $y$ -intercepts.

Assumptions1. The polynomial function is $f(x)=(x+1)(x-)(x+5)$ . We need to analyze the function in terms of its end behavior, and its $x$ and $y$ intercepts.

Next, we need to find the $x$ -intercepts of the graph of the function. The $x$ -intercepts are the values of $x$ for which $f(x)=0$ .
So, we set the function equal to zero and solve for $x$ .
$(x+1)(x-2)(x+)=0$

The equation $(x+1)(x-2)(x+5)=0$ is true if any of the factors is equal to zero. So, we set each factor equal to zero and solve for $x$ .
$x+1=0, x-2=0, x+5=0$

olving the equations $x+1=0$ , $x-2=0$ , and $x+5=0$ gives the solutions $x=-1$ , $x=2$ , and $x=-5$ .
So, the $x$ -intercepts of the graph of the function are $x=-1$ , $x=2$ , and $x=-5$ .

Finally, we need to find the $y$ -intercept of the graph of the function. The $y$ -intercept is the value of $f(x)$ when $x=0$ .
So, we substitute $x=0$ into the function and solve for $f(x)$ .
$f(0)=(0+1)(0-2)(0+5)$

Calculating $f(0)=(0+1)(0-2)(0+5)$ gives $f(0)=-10$ .
So, the $y$ -intercept of the graph of the function is $y=-10$ .