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In the function \( y = \frac{x+3}{x+2} \), a vertical asymptote occurs where the function becomes undefined. This happens when the denominator is zero. For this particular function, the denominator is \( x + 2 \), which leads us to a vertical asymptote at \( x = -2 \). This is because when \( x = -2 \), the divisor is zero, making the expression undefined.

Vertical asymptotes are important as they show where the graph will demonstrate a dramatic vertical rise or fall, essentially splitting the graph into different sections. In graphing terms, the curve will draw very close to the asymptote, but never actually touch it. On a graphing calculator, this usually shows up as a distinct break or gap in the graph's curve.

Horizontal asymptotes provide insight into the behavior of a rational function as \( x \) approaches infinity or negative infinity. For our function \( y = \frac{x+3}{x+2} \), the horizontal asymptote can be found by examining the highest power terms. Since both the numerator and denominator are first degree polynomials, divide their leading coefficients: \( \frac{1}{1} = 1 \). Thus, the horizontal asymptote is \( y = 1 \).

This means that as \( x \) becomes very large or very small, the value of \( y \) will get closer and closer to 1. While it may look like the curve is touching the line at \( y = 1 \), it actually never does. This guides us in selecting appropriate limits when setting up a graphing window, ensuring the long-term behavior is visible.

X-intercepts occur where the graph of a function crosses the x-axis. It means that at these points, \( y = 0 \). For the function \( y = \frac{x+3}{x+2} \), setting \( y \) to zero gives us the equation \( 0 = \frac{x+3}{x+2} \). Solving this, we multiply both sides by \( x+2 \) (keeping in mind we are where the function is defined):

• \( 0 = x + 3 \)
• \( x = -3 \)

So, the x-intercept is at \( x = -3 \). Plotting this accurately is key to understanding the graph's full range and shape. It's an essential feature to include in our graphing window.

The y-intercept of a function is found by evaluating it at \( x = 0 \). In the function \( y = \frac{x+3}{x+2} \), substituting \( x = 0 \) gives us \( y = \frac{0+3}{0+2} = \frac{3}{2} \). Thus, the y-intercept is \( y = \frac{3}{2} \).

The y-intercept provides a crucial reference point when plotting the function's graph. It shows where the graph crosses the y-axis. By knowing the y-intercept, we can determine more accurately the direction in which the graph will rise or fall as it moves away from the y-axis. This further aids in setting an appropriate graphing window to capture all critical features of the function.