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Writing the equation of a rational function given its graph
The figure below shows the graph of a rational function
f.
It has vertical asymptotes
x=1 and
x=5, and horizontal asymptote
y=-2.
The graph has
x-intercepts 2 and -5 , and it passes through the point
(-1,2).
The equation for
f(x) has one of the five forms shown below.
Choose the appropriate form for
f(x), and then write the equation.
You can assume that
f(x) is in simplest form.

f(x)=(a)/(x-b)
=(◻)/(◻)

f(x)=(a(x-b))/(x-c)
=(◻(◻))/(◻)
f(x)=(a)/((x-b)(x-c))=(◻)/((◻)(◻))
f(x)=(a(x-b))/((x-c)(x-d))=(◻(◻))/((◻)(◻))
f(x)=(a(x-b)(x-c))/((x-d)(x-e))=(◻(◻)(◻))/((◻)(◻))
Explanation
Check
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Writing the equation of a rational function given its graph $\newline$ The figure below shows the graph of a rational function $f$ . $\newline$ It has vertical asymptotes $x=1$ and $x=5$ , and horizontal asymptote $y=-2$ . $\newline$ The graph has $x$ -intercepts $2$ and $-5$ , and it passes through the point $(-1,2)$ . $\newline$ The equation for $f(x)$ has one of the five forms shown below. $\newline$ Choose the appropriate form for $f(x)$ , and then write the equation. $\newline$ You can assume that $f(x)$ is in simplest form. $\newline$ $f(x)=\frac{a}{x-b}$ $x=1$ $0$ $\newline$ $x=1$ $1$ $x=1$ $2$ $x=1$ $3$ $x=1$ $4$ $x=1$ $5$ $\newline$ Explanation $\newline$ Check $\newline$ ( $9$ ) $2024$ McGraw Hill LLC. All Rights Reserved. $\newline$ Terms of Use $\newline$ Privacy Center

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Find recursive and explicit formulas

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Q. Writing the equation of a rational function given its graph

\newline

The figure below shows the graph of a rational function

f

.

\newline

It has vertical asymptotes

x=1

and

x=5

, and horizontal asymptote

y=-2

.

\newline

The graph has

x

-intercepts

2

and

-5

, and it passes through the point

(-1,2)

.

\newline

The equation for

f(x)

has one of the five forms shown below.

\newline

Choose the appropriate form for

f(x)

, and then write the equation.

\newline

You can assume that

f(x)

is in simplest form.

\newline

f(x)=\frac{a}{x-b}

x=1

0

\newline

x=1

1

x=1

2

x=1

3

x=1

4

x=1

5

\newline

Explanation

\newline

Check

\newline

(

9

)

2024

McGraw Hill LLC. All Rights Reserved.

\newline

Terms of Use

\newline

Privacy Center

Identify Asymptotes: To determine the correct form of the rational function, we need to consider the given asymptotes and intercepts. Vertical asymptotes occur where the denominator of the rational function is zero, and horizontal asymptotes are determined by the degrees of the numerator and denominator.
Vertical Asymptotes: Since there are two vertical asymptotes at $x=1$ and $x=5$ , the denominator must have factors $(x-1)$ and $(x-5)$ . This means the denominator will be of the form $(x-1)(x-5)$ .
Horizontal Asymptote Determination: The horizontal asymptote is at $y=-2$ , which suggests that the degrees of the numerator and the denominator are the same, because the leading coefficients of the numerator and denominator will determine the horizontal asymptote when the degrees are equal. Therefore, the numerator must also be a quadratic expression to match the degree of the denominator.
X-Intercepts Analysis: The x-intercepts are at $x=2$ and $x=-5$ , which means the numerator must have factors $(x-2)$ and $(x+5)$ . This gives us a numerator of the form $a(x-2)(x+5)$ , where $a$ is a constant that we need to determine.
Constant Determination: The function passes through the point $(-1,2)$ . We can use this point to solve for the constant $a$ . Plugging in the point into the function, we get: $\newline$ $2 = a(-1-2)(-1+5)$ $\newline$ $2 = a(-3)(4)$ $\newline$ $2 = -12a$ $\newline$ $a = -\frac{1}{6}$
Equation Formation: Now we have all the information we need to write the equation of the function. The function has the form: $\newline$ $f(x) = \frac{a(x-2)(x+5)}{(x-1)(x-5)}$ $\newline$ Substituting the value of $a$ we found, we get: $\newline$ $f(x) = \frac{-\frac{1}{6}(x-2)(x+5)}{(x-1)(x-5)}$

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