Math Problem Statement

Write an equation for a rational function with: Vertical asymptotes at x = − 5 and x = 1 x -intercepts at ( − 4 , 0 ) and ( − 1 , 0 ) y -intercept at ( 0 , 7 )

Solution

To construct a rational function with the given properties, let’s break it down step by step.


1. General form of a rational function:

A rational function f(x)f(x) has the general form:

f(x)=N(x)D(x)f(x) = \frac{N(x)}{D(x)}

where N(x)N(x) is the numerator (containing the x-intercepts) and D(x)D(x) is the denominator (containing the vertical asymptotes). Let's build these based on the conditions.


2. Vertical asymptotes:

  • Vertical asymptotes occur when the denominator becomes zero.
  • Since the vertical asymptotes are at x=5x = -5 and x=1x = 1, the factors of the denominator must be:

D(x)=(x+5)(x1)D(x) = (x + 5)(x - 1)


3. x-intercepts:

  • The numerator determines the x-intercepts.
  • Since the x-intercepts are at (4,0)(-4, 0) and (1,0)(-1, 0), the numerator must contain these as factors:

N(x)=(x+4)(x+1)N(x) = (x + 4)(x + 1)


4. Putting it together:

So far, the rational function is:

f(x)=A(x+4)(x+1)(x+5)(x1)f(x) = \frac{A (x + 4)(x + 1)}{(x + 5)(x - 1)}

where AA is a constant that ensures the function passes through the y-intercept at (0,7)(0, 7).


5. Finding the constant AA:

  • Use the y-intercept at (0,7)(0, 7) to find AA.
  • Substituting x=0x = 0 and f(0)=7f(0) = 7:

7=A(0+4)(0+1)(0+5)(01)7 = \frac{A (0 + 4)(0 + 1)}{(0 + 5)(0 - 1)}

7=A415(1)7 = \frac{A \cdot 4 \cdot 1}{5 \cdot (-1)}

7=4A57 = \frac{-4A}{5}

35=4AA=354-35 = -4A \quad \Rightarrow \quad A = \frac{35}{4}


6. Final equation:

Thus, the rational function is:

f(x)=354(x+4)(x+1)(x+5)(x1)f(x) = \frac{\frac{35}{4} (x + 4)(x + 1)}{(x + 5)(x - 1)}

Or equivalently:

f(x)=35(x+4)(x+1)4(x+5)(x1)f(x) = \frac{35 (x + 4)(x + 1)}{4 (x + 5)(x - 1)}


Would you like further explanation or a graph of this function?


Here are 5 related questions:

  1. What happens to the function as xx \to \infty?
  2. How do horizontal asymptotes differ from vertical asymptotes?
  3. How do you find slant asymptotes for rational functions?
  4. How would the equation change if the x-intercepts had multiplicity 2?
  5. How can you determine whether the function has any holes?

Tip: When solving for constants using known points, be mindful of negative signs—small mistakes can affect the solution significantly.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Asymptotes
Intercepts

Formulas

Rational function form f(x) = N(x) / D(x)
Vertical asymptotes: set denominator factors to zero
x-intercepts: set numerator factors to zero
Substitute known point to solve for constant

Theorems

Concept of Vertical Asymptotes
Concept of Intercepts

Suitable Grade Level

Grades 11-12