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The vertex formula is an essential tool when working with quadratic functions. A quadratic function typically resembles the form \( f(x) = ax^2 + bx + c \) and its graph is a parabola. The vertex of this parabola is the point where it turns. To find the vertex's x-coordinate, we can use the formula:\[ x = \frac{-b}{2a} \]This formula gives us the x-value where the function reaches its maximum or minimum value, depending on whether the parabola opens upwards or downwards. For the function \( f(x) = 7.6x^2 + 9.8x - 2.1 \), we apply the vertex formula with \( a = 7.6 \) and \( b = 9.8 \). Thus, the x-coordinate of the vertex is \( \frac{-9.8}{2 \times 7.6} \), which calculates to approximately \(-0.64\). This critical step helps us determine at what point on the x-axis the parabola reaches its extremum (in this case, a minimum).

Understanding the vertex formula is valuable because it connects the algebraic representation of a quadratic function with its graphical interpretation. Once we have the x-coordinate, we can substitute it back into the function to find the y-coordinate, which gives the value of the function at the vertex.

A parabola is a symmetrical, U-shaped curve that is the graph of a quadratic function. Depending on the coefficient \( a \) in \( f(x) = ax^2 + bx + c \), a parabola can open upwards or downwards:

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

In the context of the given function \( f(x) = 7.6x^2 + 9.8x - 2.1 \), the coefficient \( a = 7.6 \) is positive, indicating that the parabola opens upwards. This upward-opening shape tells us that any point on the parabola can be traced back to its lowest point, known as the vertex. Since parabolas are symmetric with respect to their vertex, the line that passes through the vertex and divides the parabola into two mirrored halves is called the axis of symmetry.

Parabolas play a significant role in various fields, including physics and engineering, often modeling trajectories and optimizing solutions. Understanding how to describe and find points on a parabola equips students with powerful analytical tools for problem-solving.

The minimum value of a quadratic function occurs at the vertex if the parabola opens upwards. This is crucial because it represents the lowest point of the graph. In our solved function, \( f(x) = 7.6x^2 + 9.8x - 2.1 \), the parabola opens upwards as earlier discussed with the positive coefficient \( a \). To find this minimum value:1. Determine the x-coordinate of the vertex using \( x = \frac{-b}{2a} \). For our function, this x-value is approximately \(-0.64\).2. Substitute this x-value back into the original function to find the function's value.3. Thus, \( f(-0.64) \) gives: \[ 7.6(-0.64)^2 + 9.8(-0.64) - 2.1 \] Calculating this, we find that the minimum value is approximately \(-5.26\).Understanding the minimum value is essential for analyzing real-world problems where you're concerned with minimizing cost, distance, energy, and other quantifiable metrics that can be modeled by quadratic functions.