Example Question - rational function equation
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Equation of Rational Function with Given Conditions
To write an equation for a rational function that satisfies the given conditions, we can follow these steps: 1. Vertical asymptotes at \( x = 5 \) and \( x = -6 \) are given by factors in the denominator that become zero at these x-values. Thus, the denominator will be \( (x - 5)(x + 6) \). 2. X-intercepts at \( x = -5 \) and \( x = 1 \) indicate where the numerator of the function becomes zero. Thus, the numerator will be a product of factors that become zero at these x-values, which gives us \( (x + 5)(x - 1) \). 3. The y-intercept is the point where the function crosses the y-axis, which happens when \( x = 0 \). To get a y-intercept at 6, we need the constant term of the numerator (when \( x = 0 \)) to give us a y-value of 6. Since the denominator at \( x = 0 \) will be \( (0 - 5)(0 + 6) = -30 \), to obtain a y-intercept of 6, the numerator must evaluate to \( -30 * 6 \) when \( x = 0 \). Therefore, we can multiply the numerator by \( -6 \) to get the desired y-intercept. Putting it all together, the rational function is: \( y = \frac{-6(x + 5)(x - 1)}{(x - 5)(x + 6)} \) This function satisfies the given conditions: - The vertical asymptotes are at \( x = 5 \) and \( x = -6 \). - The x-intercepts are at \( x = -5 \) and \( x = 1 \). - The y-intercept is at \( y = 6 \) because when \( x = 0 \), \( y = \frac{-6(0 + 5)(0 - 1)}{(0 - 5)(0 + 6)} = \frac{30}{-30} = -1 \cdot -6 = 6 \).
Finding a Rational Function Given Vertical, X-intercepts, and Y-intercept
To write an equation for a rational function that meets the given requirements, you need to consider the vertical asymptotes, x-intercepts, and y-intercept provided. Vertical asymptotes occur where the denominator of the rational function is zero. Thus, for vertical asymptotes at \( x = -6 \) and \( x = 3 \), the factors in the denominator will be \( (x + 6) \) and \( (x - 3) \), since setting either of these equal to zero gives the respective values of \( x \) for the asymptotes. X-intercepts occur where the numerator is zero. Given the x-intercepts at \( x = -3 \) and \( x = -1 \), the factors in the numerator will be \( (x + 3) \) and \( (x + 1) \), since setting these factors equal to zero will yield the x-intercepts. Combining these, we get a function of the form: \[ f(x) = \frac{a(x + 3)(x + 1)}{(x + 6)(x - 3)} \] where \( a \) is a constant we need to determine using the y-intercept. The y-intercept is found by setting \( x = 0 \) in the function. It is given that the y-intercept is 10; thus, we solve for \( a \) by substituting \( x = 0 \) into the equation to get \( f(0) = 10 \): \[ 10 = \frac{a(0 + 3)(0 + 1)}{(0 + 6)(0 - 3)} \] \[ 10 = \frac{3a}{-18} \] Now solve for \( a \): \[ a = \frac{10 \times (-18)}{3} = -60 \] Insert \( a \) back into the original equation to get the complete function: \[ f(x) = \frac{-60(x + 3)(x + 1)}{(x + 6)(x - 3)} \]
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