QuestionPlot the features of the function $f(x)=\frac{x-2}{2x+7}$ : vertical asymptote, horizontal asymptote, $x$ -intercept, $y$ -intercept, and hole.

Studdy Solution

STEP 1

Assumptions1. The given rational function is $f(x)=\frac{x-}{x+7}$ . . We need to find and plot the five features of this function vertical asymptote, horizontal asymptote, $x$ -intercept, $y$ -intercept, and hole (if any).

STEP 2

First, let's find the vertical asymptote. A vertical asymptote is a vertical line $x=a$ where the function approaches $\pm\infty$ as $x$ approaches $a$ . For a rational function, the vertical asymptotes occur where the denominator is zero and the numerator is not zero.
So, we set the denominator equal to zero and solve for $x$ .
$2x+7=0$

STEP 3

olve the equation $2x+7=0$ for $x$ .
$x=-\frac{7}{2}$ So, the vertical asymptote is $x=-\frac{7}{2}$ .

STEP 4

Next, let's find the horizontal asymptote. A horizontal asymptote is a horizontal line $y=b$ where the function approaches $b$ as $x$ approaches $\pm\infty$ . For a rational function, the horizontal asymptote depends on the degrees of the numerator and denominator polynomials.
In this case, the degree of the numerator is1 and the degree of the denominator is also1. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients, which is $\frac{1}{2}$ .
So, the horizontal asymptote is $y=\frac{1}{2}$ .

STEP 5

Now, let's find the $x$ -intercept. The $x$ -intercept is the value of $x$ where the function equals zero. So, we set the function equal to zero and solve for $x$ .
$\frac{x-2}{2x+7}=0$

STEP 6

olve the equation $\frac{x-2}{2x+}=0$ for $x$ .
$x=2$ So, the $x$ -intercept is $x=2$ .

STEP 7

Next, let's find the $y$ -intercept. The $y$ -intercept is the value of the function when $x=0$ . So, we substitute $x=0$ into the function and solve for $y$ .
$f(0)=\frac{0-2}{2(0)+7}$

STEP 8

Calculate the value of $f(0)$ .
$f(0)=\frac{-2}{7}$ So, the $y$ -intercept is $y=-\frac{2}{7}$ .

STEP 9

Finally, let's check for a hole in the graph. A hole occurs where both the numerator and denominator are zero. However, in this function, there is no value of $x$ that makes both the numerator and denominator zero.
So, there is no hole in the graph of this function.
Now, we can plot these features on the graph of the function.

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QuestionPlot the features of the function $f(x)=\frac{x-2}{2x+7}$ : vertical asymptote, horizontal asymptote, $x$ -intercept, $y$ -intercept, and hole.

Studdy Solution

STEP 1

Assumptions1. The given rational function is $f(x)=\frac{x-}{x+7}$ . . We need to find and plot the five features of this function vertical asymptote, horizontal asymptote, $x$ -intercept, $y$ -intercept, and hole (if any).

STEP 2

STEP 3

olve the equation $2x+7=0$ for $x$ .
$x=-\frac{7}{2}$ So, the vertical asymptote is $x=-\frac{7}{2}$ .

STEP 4

STEP 5

Now, let's find the $x$ -intercept. The $x$ -intercept is the value of $x$ where the function equals zero. So, we set the function equal to zero and solve for $x$ .
$\frac{x-2}{2x+7}=0$

STEP 6

olve the equation $\frac{x-2}{2x+}=0$ for $x$ .
$x=2$ So, the $x$ -intercept is $x=2$ .

STEP 7

Next, let's find the $y$ -intercept. The $y$ -intercept is the value of the function when $x=0$ . So, we substitute $x=0$ into the function and solve for $y$ .
$f(0)=\frac{0-2}{2(0)+7}$

STEP 8

Calculate the value of $f(0)$ .
$f(0)=\frac{-2}{7}$ So, the $y$ -intercept is $y=-\frac{2}{7}$ .

STEP 9

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QuestionPlot the features of the function f(x)=x−22x+7f(x)=\frac{x-2}{2x+7}f(x)=2x+7x−2​: vertical asymptote, horizontal asymptote, xxx-intercept, yyy-intercept, and hole.

Studdy Solution

STEP 1

Assumptions1. The given rational function is f(x)=x−x+7f(x)=\frac{x-}{x+7}f(x)=x+7x−​. . We need to find and plot the five features of this function vertical asymptote, horizontal asymptote, xxx-intercept, yyy-intercept, and hole (if any).

STEP 2

STEP 3

olve the equation 2x+7=02x+7=02x+7=0 for xxx.x=−72x=-\frac{7}{2}x=−27​So, the vertical asymptote is x=−72x=-\frac{7}{2}x=−27​.

STEP 4

STEP 5

Now, let's find the xxx-intercept. The xxx-intercept is the value of xxx where the function equals zero. So, we set the function equal to zero and solve for xxx.x−22x+7=0\frac{x-2}{2x+7}=02x+7x−2​=0

STEP 6

olve the equation x−22x+=0\frac{x-2}{2x+}=02x+x−2​=0 for xxx.x=2x=2x=2So, the xxx-intercept is x=2x=2x=2.

STEP 7

Next, let's find the yyy-intercept. The yyy-intercept is the value of the function when x=0x=0x=0. So, we substitute x=0x=0x=0 into the function and solve for yyy.f(0)=0−22(0)+7f(0)=\frac{0-2}{2(0)+7}f(0)=2(0)+70−2​

STEP 8

Calculate the value of f(0)f(0)f(0).f(0)=−27f(0)=\frac{-2}{7}f(0)=7−2​So, the yyy-intercept is y=−27y=-\frac{2}{7}y=−72​.

STEP 9

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QuestionPlot the features of the function $f(x)=\frac{x-2}{2x+7}$ : vertical asymptote, horizontal asymptote, $x$ -intercept, $y$ -intercept, and hole.

Assumptions1. The given rational function is $f(x)=\frac{x-}{x+7}$ . . We need to find and plot the five features of this function vertical asymptote, horizontal asymptote, $x$ -intercept, $y$ -intercept, and hole (if any).

olve the equation $2x+7=0$ for $x$ .
$x=-\frac{7}{2}$ So, the vertical asymptote is $x=-\frac{7}{2}$ .

Now, let's find the $x$ -intercept. The $x$ -intercept is the value of $x$ where the function equals zero. So, we set the function equal to zero and solve for $x$ .
$\frac{x-2}{2x+7}=0$

olve the equation $\frac{x-2}{2x+}=0$ for $x$ .
$x=2$ So, the $x$ -intercept is $x=2$ .

Next, let's find the $y$ -intercept. The $y$ -intercept is the value of the function when $x=0$ . So, we substitute $x=0$ into the function and solve for $y$ .
$f(0)=\frac{0-2}{2(0)+7}$

Calculate the value of $f(0)$ .
$f(0)=\frac{-2}{7}$ So, the $y$ -intercept is $y=-\frac{2}{7}$ .