a) \( G(x)=3 x^{2}+5 x+3 \)

Did you know that quadratic functions like \( G(x) = 3x^{2} + 5x + 3 \) can be graphed as parabolas? This particular function opens upwards because the coefficient of \( x^{2} \) is positive (3). The vertex can be found using the formula \( x = -\frac{b}{2a} \), which for this function gives the x-coordinate of the vertex as \( -\frac{5}{2(3)} = -\frac{5}{6} \), giving you a nice starting point to sketch its shape! In real-world applications, quadratic functions such as this one can model various phenomena, like projectile motion or profit maximization in business. For example, if \( G(x) \) represented profit based on the number of items sold, it would help businesses find the optimal quantity to sell to maximize their profits. Understanding these concepts can provide valuable insights into practical situations!