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www-awu.aleks.com VitalSource Bookshelf: Print Reading for Construction A ALEKS - Zachary Zawacki - Learn Polynomial and Rational Functions Writing the equation of a rational function given its graph Zachary The 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. {:[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-c)),=(◻(◻)(◻))/((◻)(◻))]:} Españo

www-awu.aleks.com\newlineVitalSource Bookshelf: Print Reading for Construction\newlineA\newlineALEKS - Zachary Zawacki - Learn\newlinePolynomial and Rational Functions\newlineWriting the equation of a rational function given its graph\newlineZachary\newlineThe figure below shows the graph of a rational function f f .\newlineIt has vertical asymptotes x=−1 x=-1 and x=−5 x=-5 , and horizontal asymptote y=0 y=0 .\newlineThe graph has x x -intercept −4-4 , and it passes through the point (2,2) (2,2) .\newlineThe equation for f(x) f(x) has one of the five forms shown below.\newlineChoose the appropriate form for f(x) f(x) , and then write the equation.\newlineYou can assume that f(x) f(x) is in simplest form.\newlinef(x)=ax−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−c)=□(□)(□)(□)(□) \begin{array}{ll} f(x)=\frac{a}{x-b} & =\frac{\square}{\square} \\ f(x)=\frac{a(x-b)}{x-c} & =\frac{\square(\square)}{\square} \\ f(x)=\frac{a}{(x-b)(x-c)} & =\frac{\square}{\square(\square)} \\ f(x)=\frac{a(x-b)}{(x-c)(x-d)} & =\frac{\square(\square)}{(\square)(\square)} \\ f(x)=\frac{a(x-b)(x-c)}{(x-d)(x-c)} & =\frac{\square(\square)(\square)}{(\square)(\square)} \end{array} \newlineEspaño

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Q. www-awu.aleks.com\newlineVitalSource Bookshelf: Print Reading for Construction\newlineA\newlineALEKS - Zachary Zawacki - Learn\newlinePolynomial and Rational Functions\newlineWriting the equation of a rational function given its graph\newlineZachary\newlineThe figure below shows the graph of a rational function f f .\newlineIt has vertical asymptotes x=−1 x=-1 and x=−5 x=-5 , and horizontal asymptote y=0 y=0 .\newlineThe graph has x x -intercept −4-4 , and it passes through the point (2,2) (2,2) .\newlineThe equation for f(x) f(x) has one of the five forms shown below.\newlineChoose the appropriate form for f(x) f(x) , and then write the equation.\newlineYou can assume that f(x) f(x) is in simplest form.\newlinef(x)=ax−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−c)=□(□)(□)(□)(□) \begin{array}{ll} f(x)=\frac{a}{x-b} & =\frac{\square}{\square} \\ f(x)=\frac{a(x-b)}{x-c} & =\frac{\square(\square)}{\square} \\ f(x)=\frac{a}{(x-b)(x-c)} & =\frac{\square}{\square(\square)} \\ f(x)=\frac{a(x-b)}{(x-c)(x-d)} & =\frac{\square(\square)}{(\square)(\square)} \\ f(x)=\frac{a(x-b)(x-c)}{(x-d)(x-c)} & =\frac{\square(\square)(\square)}{(\square)(\square)} \end{array} \newlineEspaño
  1. Analyze Information: Analyze the given information about the rational function f(x)f(x). We know that f(x)f(x) has vertical asymptotes at x=−1x = -1 and x=−5x = -5, a horizontal asymptote at y=0y = 0, an xx-intercept at −4-4, and it passes through the point (2,2)(2,2). This information will help us determine the form of the function.
  2. Determine Function Form: Determine the form of the function based on the vertical asymptotes.\newlineSince there are two vertical asymptotes at x=−1x = -1 and x=−5x = -5, the denominator of the function must have factors (x+1)(x + 1) and (x+5)(x + 5). This eliminates the first two forms, as they do not have two factors in the denominator.
  3. Consider Horizontal Asymptote: Consider the horizontal asymptote. The horizontal asymptote at y=0y = 0 suggests that the degree of the numerator must be less than the degree of the denominator. This eliminates the last form, as it has a numerator of higher degree than the denominator.
  4. Use X-Intercept: Use the x-intercept to further refine the form of the function.\newlineThe x-intercept at −4-4 suggests that the numerator must have a factor of (x+4)(x + 4). This eliminates the third form, as it does not have a factor in the numerator.
  5. Write Function with Given Point: Write the function using the remaining form and the given point.\newlineThe remaining form is f(x)=a(x−b)(x−c)(x−d)f(x) = \frac{a(x - b)}{(x - c)(x - d)}, which becomes f(x)=a(x+4)(x+1)(x+5)f(x) = \frac{a(x + 4)}{(x + 1)(x + 5)}. We need to find the value of aa using the point (2,2)(2,2).
  6. Solve for aa: Plug in the point (2,2)(2,2) to solve for aa.2=a(2+4)(2+1)(2+5)2 = \frac{a(2 + 4)}{(2 + 1)(2 + 5)}2=6a3×72 = \frac{6a}{3 \times 7}2=6a212 = \frac{6a}{21}42=6a42 = 6aa=7a = 7
  7. Final Function Equation: Write the final equation of the function.\newlineThe final equation of the function is f(x)=7(x+4)(x+1)(x+5)f(x) = \frac{7(x + 4)}{(x + 1)(x + 5)}.

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