1. **State the problem:** We are given the function $$y = \frac{2x + 1}{x - 4}$$ and we want to understand its behavior. 2. **Formula and rules:** This is a rational function, which is a ratio of two polynomials. Important rules: - The denominator cannot be zero because division by zero is undefined. - The function may have vertical asymptotes where the denominator is zero. - The function may have horizontal or oblique asymptotes depending on the degrees of numerator and denominator. 3. **Find domain restrictions:** Set denominator equal to zero: $$x - 4 = 0$$ $$x = 4$$ So, $$x = 4$$ is not in the domain. 4. **Simplify and analyze:** The function is already simplified. 5. **Find vertical asymptote:** At $$x = 4$$ because denominator is zero. 6. **Find horizontal asymptote:** Compare degrees of numerator and denominator (both degree 1). The horizontal asymptote is the ratio of leading coefficients: $$y = \frac{2}{1} = 2$$ 7. **Find intercepts:** - **y-intercept:** Set $$x=0$$: $$y = \frac{2(0) + 1}{0 - 4} = \frac{1}{-4} = -\frac{1}{4}$$ - **x-intercept:** Set $$y=0$$, numerator must be zero: $$2x + 1 = 0$$ $$2x = -1$$ $$x = -\frac{1}{2}$$ **Final answer:** The function $$y = \frac{2x + 1}{x - 4}$$ has a vertical asymptote at $$x=4$$, a horizontal asymptote at $$y=2$$, a y-intercept at $$\left(0, -\frac{1}{4}\right)$$, and an x-intercept at $$\left(-\frac{1}{2}, 0\right)$$.