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The vertex of a quadratic function is a crucial concept in understanding its graphical representation. A quadratic function is typically written in the form \( ax^2 + bx + c \). The vertex is either the highest or lowest point on its parabola depending on the parabola's direction. This direction is determined by the coefficient \(a\):

• If \(a < 0\), the parabola opens downward, and the vertex is the maximum point.
• If \(a > 0\), the parabola opens upward, and the vertex is the minimum point.

For the quadratic function \(F(t) = -0.0076 t^2 + 0.45 t + 16\), we find the vertex's x-coordinate using the formula \( t = -\frac{b}{2a} \). By substituting \(a = -0.0076\) and \(b = 0.45\), we find:\[t = -\frac{0.45}{2(-0.0076)} \approx 29.61\]This calculation reveals the theoretical time when fuel economy might peak. However, due to the constraint \(0 \leq t \leq 28\), we adjust \(t\) to its practical maximum within the context, which is 28.

The concept of fuel economy is an essential aspect of vehicle efficiency studies. It measures how far a vehicle can travel per unit of fuel, often expressed in miles per gallon (mpg). The quadratic function in our problem, \(F(t) = -0.0076 t^2 + 0.45 t + 16\), models fuel economy over time for passenger cars in the U.S. from 1980 onward.

• "\(t\)" represents years since 1980, making it straightforward to align the model with specific calendar years. For instance, \(t = 10\) corresponds to 1990.
• Fuel economy changes modelled by this function reflect technological improvements, regulatory changes, or fluctuations in fuel standards.

Interpreting the vertex \((28, 23.24)\) of this function suggests that in the year 2008, the average fuel economy peaked at about 23.24 mpg. This information is useful for transportation analysts and policymakers working to improve fuel efficiency.

Understanding how to analyze a parabola graph is integral for interpreting quadratic functions effectively. A parabola, the graph of a quadratic equation, is symmetrical and its shape is governed by the leading coefficient \(a\). The vertex is at the center of this symmetry. Here's an approach to analyzing parabolas:

• Identify the vertex to locate either the maximum or minimum point of the function.
• Look at the axis of symmetry, which for the function \(y = ax^2 + bx + c\) is a vertical line passing through the vertex \(x = -\frac{b}{2a}\).
• Examine the "b" and "c" values — they influence the parabola's direction and vertical shift respectively.

For the quadratic function modeling fuel economy \(F(t) = -0.0076 t^2 + 0.45 t + 16\), the parabola opens downward (as \(a < 0\)), which means the vertex at \(t = 28\) is the peak. This aligns with understanding that cars introduced new technologies by 2008, achieving significant fuel economy before potentially plateauing. An analysis like this informs both economic forecasts and engineering planning.