1)) Find the square. Simplify your answer. \( (y+1)^{2} \)

To find the square of \( (y+1)^{2} \), you can use the formula for squaring a binomial, which is \( (a+b)^{2} = a^{2} + 2ab + b^{2} \). Here, \( a = y \) and \( b = 1 \). So, \( (y+1)^{2} = y^{2} + 2(y)(1) + 1^{2} = y^{2} + 2y + 1 \). The simplified answer is \( y^{2} + 2y + 1 \). Now let's explore this a bit! This expression \( (y+1)^{2} \) represents a perfect square trinomial, which not only makes it neat but also reveals that its graph is a parabola opening upwards with its vertex at (-1, 0). Snazzy, right? When applying this in real-world scenarios, think about situations like calculating the area of a square plot of land where the side length is represented by \( (y+1) \). Expressing it as \( y^{2} + 2y + 1 \) allows you to break down and analyze how changes in \( y \) affect the area almost intuitively, making it a handy mathematical tool!