Math Snap

STEP 1

1. The function is a rational function with vertical asymptotes at $x = -3$ and $x = 4$.
2. The horizontal asymptote is at $y = -1$.
3. The $x$-intercepts are at $(5, 0)$ and $(-4, 0)$.
4. The $y$-intercept is at $\left(0, -\frac{5}{3}\right)$.

STEP 2

1. Determine the form of the rational function based on asymptotes.
2. Incorporate $x$-intercepts into the function.
3. Use the $y$-intercept to find the constant factor.
4. Write the final reduced rational function.

STEP 3

The vertical asymptotes at $x = -3$ and $x = 4$ suggest that the denominator of the rational function has factors $(x + 3)$ and $(x - 4)$.
The horizontal asymptote at $y = -1$ indicates that the degrees of the numerator and denominator are the same, and the leading coefficient of the numerator divided by the leading coefficient of the denominator is $-1$.

STEP 4

The $x$-intercepts at $(5, 0)$ and $(-4, 0)$ suggest that the numerator has factors $(x - 5)$ and $(x + 4)$.

STEP 5

The rational function can be expressed as:
$$f(x) = \frac{a(x - 5)(x + 4)}{(x + 3)(x - 4)}$$ To find $a$, use the $y$-intercept $\left(0, -\frac{5}{3}\right)$:
$$f(0) = \frac{a(0 - 5)(0 + 4)}{(0 + 3)(0 - 4)} = -\frac{5}{3}$$ $$\frac{a(-5)(4)}{3(-4)} = -\frac{5}{3}$$ $$\frac{-20a}{-12} = -\frac{5}{3}$$ $$\frac{5a}{3} = -\frac{5}{3}$$ $$5a = -5$$ $$a = -1$$

Substitute $a = -1$ back into the function:
$$f(x) = \frac{-(x - 5)(x + 4)}{(x + 3)(x - 4)}$$ Simplify the numerator:
$$f(x) = \frac{-(x^2 - x - 20)}{(x + 3)(x - 4)}$$ $$f(x) = \frac{-x^2 + x + 20}{(x + 3)(x - 4)}$$ This is the reduced rational function.
The possible formula for the function is:
$$f(x) = \frac{-x^2 + x + 20}{(x + 3)(x - 4)}$$