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www-awu.aleks.com\newlineVitalSource Bookshelf: Print Reading for C\'onstruction\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 ff. It has vertical asymptotes x=1x=-1 and x=5x=-5, and horizontal asymptote y=0y=0. The graph has xx-intercept 4-4, and it passes through the point (2,2)(2,2). The equation for f(x)f(x) has one of the five forms shown below. Choose the appropriate form for f(x)f(x), and then write the equation. You can assume that f(x)f(x) is in simplest form.\newline\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}\newlineEspa\~no

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Q. www-awu.aleks.com\newlineVitalSource Bookshelf: Print Reading for C\'onstruction\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 ff. It has vertical asymptotes x=1x=-1 and x=5x=-5, and horizontal asymptote y=0y=0. The graph has xx-intercept 4-4, and it passes through the point (2,2)(2,2). The equation for f(x)f(x) has one of the five forms shown below. Choose the appropriate form for f(x)f(x), and then write the equation. You can assume that f(x)f(x) is in simplest form.\newline\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}\newlineEspa\~no
  1. Identify Function Criteria: Identify the correct form of the rational function based on the given asymptotes and intercepts.\newlineSince the graph has vertical asymptotes at x=1x=-1 and x=5x=-5, the denominator of the rational function must have factors (x+1)(x+1) and (x+5)(x+5). The horizontal asymptote at y=0y=0 suggests that the degree of the numerator is less than the degree of the denominator. The xx-intercept at 4-4 means the numerator must have a factor of (x+4)(x+4).
  2. Choose Correct Form: Choose the correct form from the given options.\newlineThe form that fits the criteria from Step 11 is f(x)=a(xb)(xc)(xd)f(x)=\frac{a(x-b)}{(x-c)(x-d)}, where bb is the xx-intercept and cc, dd are the vertical asymptotes.
  3. Plug in Values: Plug in the known values for the intercepts and asymptotes into the chosen form.\newlineThe equation becomes f(x)=a(x+4)(x+1)(x+5)f(x)=\frac{a(x+4)}{(x+1)(x+5)}.
  4. Solve for Constant: Use the point (2,2)(2,2) to solve for the constant aa. Plugging in the point (2,2)(2,2) into the equation gives us 2=a(2+4)(2+1)(2+5)2 = \frac{a(2+4)}{(2+1)(2+5)}. Simplifying, we get 2=6a3×72 = \frac{6a}{3\times7}, which simplifies to 2=6a212 = \frac{6a}{21}. Multiplying both sides by 2121 gives us 42=6a42 = 6a, and dividing by 66 gives us a=7a = 7.
  5. Write Final Equation: Write the final equation of the rational function using the value of aa. The final equation is f(x)=7(x+4)(x+1)(x+5)f(x)=\frac{7(x+4)}{(x+1)(x+5)}.

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