www-awu.aleks.comVitalSource Bookshelf: Print Reading for C\'onstructionAALEKS - Zachary Zawacki - LearnPolynomial and Rational FunctionsWriting the equation of a rational function given its graphZacharyThe figure below shows the graph of a rational function f. It has vertical asymptotes x=−1 and x=−5, and horizontal asymptotey=0. The graph has x-intercept −4, and it passes through the point (2,2). 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.\begin{array}{l}
f(x)=\frac{a}{x-b}=\left(\square\right)/\left(\square\right)\
f(x)=\frac{a(x-b)}{x-c}=\left(\square(\square)\right)/\left(\square\right)\
f(x)=\frac{a}{(x-b)(x-c)}=\left(\square\right)/\left(\square(\square)\right)\
f(x)=\frac{a(x-b)}{(x-c)(x-d)}=\left(\square(\square)\right)/\left(\square\right)\
f(x)=\frac{a(x-b)(x-c)}{(x-d)(x-e)}=\left(\square(\square)(\square)\right)/\left(\square\right)\left(\square\right)
\end{array}Espa\~no
Q. www-awu.aleks.comVitalSource Bookshelf: Print Reading for C\'onstructionAALEKS - Zachary Zawacki - LearnPolynomial and Rational FunctionsWriting the equation of a rational function given its graphZacharyThe figure below shows the graph of a rational function f. It has vertical asymptotes x=−1 and x=−5, and horizontal asymptote y=0. The graph has x-intercept −4, and it passes through the point (2,2). 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.\begin{array}{l}
f(x)=\frac{a}{x-b}=\left(\square\right)/\left(\square\right)\
f(x)=\frac{a(x-b)}{x-c}=\left(\square(\square)\right)/\left(\square\right)\
f(x)=\frac{a}{(x-b)(x-c)}=\left(\square\right)/\left(\square(\square)\right)\
f(x)=\frac{a(x-b)}{(x-c)(x-d)}=\left(\square(\square)\right)/\left(\square\right)\
f(x)=\frac{a(x-b)(x-c)}{(x-d)(x-e)}=\left(\square(\square)(\square)\right)/\left(\square\right)\left(\square\right)
\end{array}Espa\~no
Identify Function Criteria: Identify the correct form of the rational function based on the given asymptotes and intercepts.Since the graph has vertical asymptotes at x=−1 and x=−5, the denominator of the rational function must have factors(x+1) and (x+5). The horizontal asymptote at y=0 suggests that the degree of the numerator is less than the degree of the denominator. The x-intercept at −4 means the numerator must have a factor of (x+4).
Choose Correct Form: Choose the correct form from the given options.The form that fits the criteria from Step 1 is f(x)=(x−c)(x−d)a(x−b), where b is the x-intercept and c, d are the vertical asymptotes.
Plug in Values: Plug in the known values for the intercepts and asymptotes into the chosen form.The equation becomes f(x)=(x+1)(x+5)a(x+4).
Solve for Constant: Use the point (2,2) to solve for the constant a. Plugging in the point (2,2) into the equation gives us 2=(2+1)(2+5)a(2+4). Simplifying, we get 2=3×76a, which simplifies to 2=216a. Multiplying both sides by 21 gives us 42=6a, and dividing by 6 gives us a=7.
Write Final Equation: Write the final equation of the rational function using the value of a. The final equation is f(x)=(x+1)(x+5)7(x+4).
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