91Ó°ÊÓ

Quadratic functions are a fundamental component of algebra and calculus, often represented by the general form: \( ax^2 + bx + c \). In the given problem, we are dealing with the function \( f(x) = -x^2 + 10x \). Here:

• The coefficient \( -1 \) in front of \( x^2 \) suggests the parabola opens downwards.
• 10 is the coefficient of \( x \), which affects the slope of the curve.
• The constant term is not present, or you could view it as 0.

The shape of a quadratic function is a parabola. In this case, our parabola will reach its maximum point because it opens downwards. The vertex of a downward opening parabola is the highest point.
This concept is important when interpreting function behavior within specific intervals. Quadratic functions can model various real-world processes, such as projectile motion or area calculations. Understanding the nature of the parabola helps predict the function's values over any interval.

Intervals are crucial in analyzing functions because they define the scope over which you are evaluating the function. In the exercise, we deal with two intervals: (0, 5) and (5, 10).

• An interval is a range of values, usually expressed in parentheses or brackets.
• The open interval (a, b) includes all numbers between a and b but not a or b themselves.
• Intervals can be open or closed; parentheses indicate open, while square brackets denote closed.

In our problem, both intervals are open which means we're looking at values very near to but not including 0, 5, or 10.
These intervals are boundaries for calculating the function's average rate of change. Understanding this allows you to deduce how the function behaves between any two points. For students, intervals form a foundation for interpreting data across different domains.

Function evaluation is the process of finding the output of a function for given inputs. It involves substituting specific values into the function equation.
Here’s a breakdown of how we do it in the problem:

• Start with the function: \( f(x) = -x^2 + 10x \).
• Evaluate \( f(x) \) at each endpoint for the given intervals.
• For (0, 5): Calculate \( f(0) \) and \( f(5) \).
• For (5, 10): Calculate \( f(5) \) and \( f(10) \).

To evaluate \( f(0) \): Substitute 0 in for \( x \), giving \( f(0) = -0^2 + 10 \times 0 = 0 \).
For \( f(5) \): Substitute 5 in for \( x \), giving \( f(5) = -(5)^2 + 10 \times 5 = -25 + 50 = 25 \).
Repeat similarly for other values. Function evaluation is a critical skill, allowing precise calculation of function values essential for deeper analysis, like rate of change or real-world applications.