91Ó°ÊÓ

Asymptotes are lines that a graph approaches but never actually reaches. They are crucial in understanding the behavior of rational functions. In the given function, \( y = \frac{1}{(x+1)^2} \), we have both a vertical and a horizontal asymptote.

- **Vertical Asymptote**: This occurs at any value of \( x \) that makes the denominator equal to zero. For this function, setting \( x+1 = 0 \) gives us a vertical asymptote at \( x = -1 \). This means as \( x \) gets closer to \(-1\), the value of \( y \) grows very large, either positively or negatively.- **Horizontal Asymptote**: Horizontal asymptotes describe the behavior of a graph as \( x \) moves towards positive or negative infinity. In this function, as \( x \to \pm \infty \), the value of \( y \) gets closer and closer to zero without ever reaching it. Thus, the x-axis \( (y = 0) \) becomes a horizontal asymptote.

These asymptotes help us sketch the overall shape of the graph: it rises sharply near the vertical asymptote and flattens out along the horizontal asymptote as \( x \) moves away from the asymptote.

The "domain" of a function refers to all possible inputs (\( x \) values), while the "range" indicates all possible outputs (\( y \) values). For rational functions, determining the domain often involves finding any points where the denominator is zero, as the function is undefined at these points.

For the function \( y = \frac{1}{(x+1)^2} \), the domain is all real numbers except for \( x = -1 \). This is because plugging \( x = -1 \) into the denominator results in division by zero.

• Domain: \( x \in \mathbb{R}, \ x eq -1 \)

Similarly, the range is determined by considering the possible \( y \) values as \( x \) shifts within the domain. For this function:

• Range: \( y > 0 \)

The output is always positive, as the division of a positive number by a non-zero positive squared value cannot yield a negative or zero result.

Key points are specific coordinates that help you sketch how the graph behaves around certain important sections of the function. Identifying these points provides a clearer picture before plotting the actual graph. For \( y = \frac{1}{(x+1)^2} \), computation of some key points involves substituting selected values into the equation:

• For \( x = 0 \), \( y = 1 \); point (0, 1)
• For \( x = 1 \), \( y = \frac{1}{4} \); point (1, 0.25)
• For \( x = -2 \), \( y = 1 \); point (-2, 1)
• For \( x = -3 \), \( y = \frac{1}{4} \); point (-3, 0.25)

These points provide a solid basis for sketching the curve. As the graph is drawn, it should smoothly pass through these points. When near \( x = -1 \), observe the steep increase in \( y \) values, climbing towards the vertical asymptote, and then note how the curve flattens out toward the horizontal asymptote when further away from the asymptote.