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A parabola is a curve that you often see in mathematics, particularly when dealing with quadratic functions. The graph of any quadratic function such as \( ax^2 + bx + c \) takes the form of a parabola. Parabolas can either open upward like a U or downward like an inverted U depending on the coefficient \( a \):

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards.

In our exercise, the quadratic function is \( f(t) = 10t^2 + 40t + 113 \). Here, \( a = 10 \), which is positive. Therefore, the parabola opens upwards.
An upward opening parabola indicates the function has a minimum point, which is important for finding minimum values. Knowing whether a parabola opens upward or downward gives insight into the limits of the values that the quadratic function can achieve.

The vertex of a parabola is crucial because it gives us the highest or lowest point of the curve, depending on the direction the parabola opens. For the quadratic function \( ax^2 + bx + c \), the formula to find the vertex's \( t \)-coordinate is:

• \( t = -\frac{b}{2a} \)

The vertex formula provides a straightforward method to determine the point where the function achieves its maximum or minimum value.
In the given function, \( f(t) = 10t^2 + 40t + 113 \), we have \( a = 10 \) and \( b = 40 \). Plugging these values into our formula, we find:

• \( t = -\frac{40}{2 \times 10} = -2 \)

This calculation shows that the vertex of the parabola occurs at \( t = -2 \). Now, by finding \( f(t) \) at this point, we can determine the minimum value of the function.

Since our parabola opens upwards, the vertex represents the minimum point of the quadratic function. Once we have calculated the \( t \)-coordinate of the vertex, the next step is to find the function value at this point for the minimum value:

• Substitute \( t = -2 \) into the function: \( f(-2) = 10(-2)^2 + 40(-2) + 113 \)

Let's break it down:

• Calculate \((-2)^2 = 4\)
• Then, \(10 \times 4 = 40\)
• Next, \(40 \times (-2) = -80\)
• Finally, substitute into the expression: \( 40 - 80 + 113 \)

By computing the above steps, you get:

• \( 40 - 80 = -40 \)
• \( -40 + 113 = 73 \)

Thus, the minimum value of the function \( f(t) \) is \( 73 \) at \( t = -2 \). This point gives the lowest value of the parabola and helps in understanding the function's behavior.