91Ó°ÊÓ

A quadratic function is a polynomial of degree 2, with a general form of \( f(t) = at^2 + bt + c \). These functions produce graphs that have a specific U-shaped curve known as a parabola. In the given exercise, our quadratic function is \( f(t) = t^2 - t \), with \( a = 1 \), \( b = -1 \), and \( c = 0 \). This function is commonly used in various mathematical modeling scenarios due to its speed of simplicity and efficiency in solving optimization problems.

• Quadratic Coefficient \( (a) \): Determines the direction in which the parabola opens; upwards if \( a > 0 \) and downwards if \( a < 0 \).
• Linear Coefficient \( (b) \): Affects the tilt or walk of the parabola along the t-axis.
• Constant Term \( (c) \): Determines where the parabola intersects the t-axis.

For \( f(t) = t^2 - t \), the parabola opens upwards as \( a = 1 \), an essential trait to remember for graphing.

A definite integral is used to calculate the area under a curve within a specific interval. In our scenario, it helps us find the average value of the quadratic function \( f(t) = t^2 - t \) over the interval \([-2, 1]\). For any function \( f(t) \), the average value over the interval \([a, b]\) can be found using the formula:\[ f_{ave} = \frac{1}{b-a} \int_a^b f(t) \, dt \]This formula allows us to integrate \( f(t) = t^2 - t \), obtaining an area value, which is then adjusted by the interval length \( (b-a) \) to give us the average. Calculating the definite integral first involves finding the antiderivative, which is then evaluated from \( a \) to \( b \). This process ensures we compute the exact area connected to the curve for our function.

A parabola is the graphical representation of a quadratic function and has a distinct U-shape. The parabola can open upward or downward depending on the sign of the coefficient of the quadratic term \( t^2 \). In our function, \( f(t) = t^2 - t \), the parabola opens upwards, as the coefficient of \( t^2 \) is positive.To better understand parabolas, it's essential to know:

• Vertex: The peak or the lowest point of a parabola, which can be calculated using \( t = -\frac{b}{2a} \). For this function, the vertex is at \( t = 0.5 \).
• Axis of Symmetry: The line that runs vertically through the vertex, dividing the parabola into two mirror-image halves. It reflects the symmetry all parabolas inherently showcase due to their geometric nature.
• Roots/Intercepts: Points where the parabola crosses the t-axis. For \( f(t) = t^2 - t \), roots occur at \( t = 0 \) and \( t = 1 \).

Understanding these features allows one to sketch the parabola more precisely and appreciate its symmetry and shape.

An interval indicates a specific range on the t-axis where we analyze or compute properties of a function. The interval used in this problem is \([-2, 1]\), which means our computations are limited to this part of the domain.When working with intervals, it's vital to remember:

• The interval \([-2, 1]\) denotes all values between -2 and 1, inclusive, meaning \(-2\) and \(1\) are part of the set.
• We use intervals to focus our study on certain pieces of a graph, finding relevant features like the average value across that range.
• The endpoints of the interval determine the "a" and "b" values in integral calculations. For this function, \( a = -2 \) and \( b = 1 \).

Being able to understand and utilize intervals efficiently simplifies the process of analyzing functions, especially when constructing graphs and evaluating average values.