91Ó°ÊÓ

A quadratic function is a type of polynomial that involves a variable raised to the second power, often in the form of \( ax^2 + bx + c \). In our exercise, we have the quadratic function \( f(x) = -x^2 + x \). This means that:

• The leading term is \(-x^2\), which indicates that the parabola opens downward because of the negative sign.
• The next coefficient is \(x\), representing the linear component of the function.

Quadratic functions like \( f(x) \) have a parabolic graph. They are symmetric about a vertical axis that passes through its vertex—the highest or lowest point depending on the parabola's orientation.
To evaluate this function, you substitute the variable with a specific number. For example, substituting \( x = 5 \) into \( f(x) = -x^2 + x \) results in \(-5^2 + 5 = -25 + 5 = -20\). However, errors were made in the solution notes, and it should be calculated as \(-25 + 5 = -20\), not \(-18\), if calculated independently from the given steps.

Linear functions are the simplest kind of polynomial functions. They are usually in the form \( ax + b \), where \( a \) and \( b \) are constants. In this exercise, the linear function given is \( h(x) = -2x + 1 \). Here's what it involves:

• The slope, \(-2\), indicates how steeply the line rises or falls; a negative slope means the line falls as \( x \) increases.
• The y-intercept, \(1\), is where the line crosses the y-axis, showing the value of the function when \( x = 0 \).

Evaluating a linear function is simple: plug the number for \( x \) into the expression. In this problem, with \( x = -2 \), substitution into \( h(x) \) yields \(-2(-2) + 1 = 4 + 1 = 5\). Therefore, \( h(-2) \) evaluates to \(5\).
Linear functions produce straight lines on a graph, making them visually easy to understand and calculate.

Function evaluation involves finding the output of a function for a particular input. It's like solving an equation where you substitute the input into the function expression and solve to find the result. In our steps, we evaluated a composition of functions, \( f(h(-2)) \). Here's how that process works:

• First, evaluate \( h(x) \) with an input of \(-2\). We found \( h(-2) = 5 \).
• Next, take that result and use it as the input for \( f(x) \). Substitute \( 5 \) into \( f(x) = -x^2 + x \) yielding \(-5^2 + 5 = -25 + 5 = -20\).
• Make sure to perform calculations step-by-step to avoid errors.

This process of substituting one function's output into another is called function composition. It allows for more complex relationships and calculations within mathematics, serving as a powerful tool for understanding dynamic systems.