Problem 506 - Precalculus

Question

For the following exercises, write an equation for a rational function with the given characteristics. Vertical asymptotes at \(x=-4\) and \(x=-1, x\) -intercepts at \((1,0)\) and \((5,0), y\) -intercept at \((0,7)\)

Step-by-Step Solution

Answer

The rational function is \(f(x) = \frac{28(x - 1)(x - 5)}{5(x + 4)(x + 1)}\).

Step 1: Understand the Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero. Given vertical asymptotes at \(x = -4\) and \(x = -1\), the denominator of our rational function will have factors \((x + 4)\) and \((x + 1)\). Thus, the denominator of our function will be \((x + 4)(x + 1)\).

Step 2: Identify the x-Intercepts

The x-intercepts of a rational function occur where the numerator equals zero. Given x-intercepts at \((1, 0)\) and \((5, 0)\), the numerator will have factors \((x - 1)\) and \((x - 5)\). Thus, the numerator of our function will be \((x - 1)(x - 5)\).

Step 3: Consider the y-Intercept

The y-intercept of a rational function is found by evaluating the function at \(x = 0\). Given a y-intercept at \((0, 7)\), we substitute \(0\) in the function and adjust the constant, if necessary, to ensure the function meets this condition.

Step 4: Formulating the Equation

Combining all the factors, we have the initial rational function form as \(f(x) = \frac{(x - 1)(x - 5)}{(x + 4)(x + 1)}\). Evaluate this function at \(x = 0\): \[f(0) = \frac{(0 - 1)(0 - 5)}{(0 + 4)(0 + 1)} = \frac{5}{4}\]Since we need the y-intercept to be \(7\), multiply the entire function by \(c\) such that \(c \cdot \frac{5}{4} = 7\).

Step 5: Solve for the Constant

Solve for \(c\) using \(c \cdot \frac{5}{4} = 7\). Multiply both sides by \(\frac{4}{5}\) to get \[c = 7 \cdot \frac{4}{5} = \frac{28}{5}\].

Step 6: Write the Final Equation

Substitute \(c = \frac{28}{5}\) back into the function, giving \[f(x) = \frac{28(x - 1)(x - 5)}{5(x + 4)(x + 1)}\]. This is the required rational function with the specified characteristics.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines where the graph of a rational function approaches but never touches. These occur where the denominator of the function equals zero since division by zero is undefined. For a rational function, these are crucial features because they highlight where the function experiences infinite behavior. Given vertical asymptotes at \(x = -4\) and \(x = -1\), the denominator must include the factors that become zero at these points:

• For \(x = -4\), the factor is \((x + 4)\) because \((x + 4) = 0\) implies \(x = -4\).


• For \(x = -1\), the factor is \((x + 1)\) since \((x + 1) = 0\) gives \(x = -1\).

Thus, our denominator is \((x + 4)(x + 1)\). Each of these factors causes the function's value to increase without bound as \(x\) approaches those values, explaining the existence of vertical asymptotes at those points.

X-Intercepts

X-intercepts are points where the graph of the function crosses the x-axis. For a rational function, these occur where the numerator equals zero. This is because a function equals zero only when its numerator is zero while the denominator is not zero.

• Given x-intercepts at \((1,0)\) and \((5,0)\), the numerator must include factors that become zero at these specific points.


• For the intercept \((1, 0)\), the factor is \((x - 1)\) because \((x - 1) = 0\) when \(x = 1\).


• Similarly, for the intercept \((5, 0)\), the factor is \((x - 5)\) since \((x - 5) = 0\) when \(x = 5\).

As a result, the numerator of our rational function is \((x - 1)(x - 5)\). These factors ensure that the function has roots at the specified x-intercepts.

Y-Intercepts

The y-intercept of a function represents where the graph crosses the y-axis, which occurs when \(x = 0\). This point offers insight into the behavior of the function as it intersects the vertical axis.

• In this exercise, the y-intercept is \((0, 7)\), meaning when \(x = 0\), \(f(x) = 7\).


• To find the y-intercept of a rational function, the value of the function is evaluated at \(x = 0\).


• Initially, substitute \(x = 0\) in the function \(f(x) = \frac{(x - 1)(x - 5)}{(x + 4)(x + 1)}\), giving \(f(0) = \frac{(-1)(-5)}{(4)(1)} = \frac{5}{4}\).

Since we desire the y-intercept to be \(7\), adjust the function by introducing a constant factor \(c\) so that \(c \cdot \frac{5}{4} = 7\). Solve this equation for \(c\) to discover the correct constant that ensures the function passes through the y-intercept \((0, 7)\). This adjustment finalizes the specific vertical position of the function relative to its y-intercept.