Graph all asymptotes of the rational function. \[ f(x)=\frac{4 x^{2}+2 x-7}{2 x+3} \]

To graph all asymptotes of the rational function \( f(x) = \frac{4x^2 + 2x - 7}{2x + 3} \), we need to find both vertical and horizontal asymptotes. 1. **Vertical Asymptotes**: Vertical asymptotes occur where the denominator is zero, provided that the numerator is not also zero at those points. Thus, we first set the denominator equal to zero: \[ 2x + 3 = 0 \] Solving for \( x \), we get: \[ 2x = -3 \quad \Rightarrow \quad x = -\frac{3}{2} \] Therefore, there is a vertical asymptote at \( x = -\frac{3}{2} \). 2. **Horizontal Asymptotes**: Horizontal asymptotes are determined by comparing the degrees of the polynomial in the numerator and the polynomial in the denominator. Here, the degree of the numerator (which is 2) is greater than the degree of the denominator (which is 1). When the degree of the numerator is greater than the degree of the denominator, the horizontal asymptote is at \( y = 0 \). In summary, the asymptotes for the function \( f(x) = \frac{4x^2 + 2x - 7}{2x + 3} \) are: - Vertical asymptote at \( x = -\frac{3}{2} \) - Horizontal asymptote at \( y = 0 \) To visualize these asymptotes on a graph, you would draw a dashed line for the vertical asymptote at \( x = -\frac{3}{2} \) and another dashed line along the x-axis (where \( y = 0 \)) for the horizontal asymptote.