\( f(x)=3 x^{2}+5 \) a. The minimum value is 5 . The domain is all real numbers and the range is \( y \geq 5 \). The function is decreasing to the left of \( x=0 \) and increasing to the right of \( x=0 \). b. The maximum value is 5 . The domain is all real numbers and the range is \( y \leq 5 \). The function is increasing to the left of \( x=0 \) and decreasing to the right of \( x=0 \). c. The minimum value is -5 . The domain is all real numbers and the range is \( y \geq-5 \). T function is decreasing to the left of \( x=0 \) and increasing to the right of \( x=0 \). d. The maximum value is -5 . The domain is all real numbers and the range is \( y \leq-5 \). function is increasing to the left of \( x=0 \) and decreasing to the right of \( x=0 \).

The function \( f(x) = 3x^{2} + 5 \) is a classic example of a quadratic function, where the parabola opens upwards since the coefficient of \( x^{2} \) is positive. The vertex, located at \( (0, 5) \), gives the minimum value of 5. This means that the lowest point on the graph is at that vertex, and the function extends infinitely upwards from there, confirming that the range is \( y \geq 5 \). When applying quadratic functions like this in real-world scenarios, they can model various phenomena, such as projectile motion. For instance, if you launch an object upward, the height of the object over time can often be represented by a quadratic equation, which helps predict its maximum height and the time it takes to reach that height. This makes understanding the properties of quadratics essential for fields ranging from physics to engineering!