91Ó°ÊÓ

A polynomial function is a mathematical expression that involves a sum of powers of variables, each multiplied by a coefficient. The function provided in the original exercise is \( f(x) = 2 - x^2 \). This particular function is a type of polynomial known as a quadratic function, where the highest exponent of the variable \( x \) is 2.

Polynomial functions can have terms that include different powers of \( x \). For instance, a complete polynomial function may look like \( a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients and \( n \) is a non-negative integer.

• The degree of a polynomial is determined by the highest power of \( x \) with a non-zero coefficient.
• In the function \( f(x) = 2 - x^2 \), the degree is 2, making it a quadratic function.
• Polynomial functions are continuous and smooth, which makes them predictable and great for modeling a wide range of scenarios.

Understanding how polynomial functions behave provides valuable insight into a wide variety of mathematical contexts.

Function substitution is a method used to evaluate a function at given points or to evaluate function expressions. It involves replacing the variable \( x \) in the function with a specific value or expression. Let's see how it applies to the current exercise.

For example, in the step "Evaluate \( f(3) \)," we substitute \( x = 3 \) into \( f(x) = 2 - x^2 \) to compute \( f(3) = 2 - 3^2 = -7 \).

With function substitution, we can find the value of an expression for any input, including:

• Simple values, as in \( f(3) \) and \( f(-1) \).
• Expressions, such as \( f(4x) \) or \( f(x-4) \).
• More complex transformations like \( f(x^2) \).

Substitution is a fundamental technique in calculus and algebra, laying the groundwork for more advanced applications like finding the derivative or integral of a function.

Function transformation refers to the alteration of a function's graph through operations like shifting, stretching, or compressing. Each transformation changes the appearance of the graph without affecting its core form. Let's break down some transformations used in your exercise:

1. **Shifts**: Shifting moves the graph horizontally or vertically.
- Horizontal shifts, like \( f(x-4) \), involve moving the graph left or right. Here, replacing \( x \) with \( x-4 \) shifts the graph 4 units to the right.
- For vertical shifts, as in \( f(x) - 4 \), subtracting 4 moves the graph down by 4 units.

2. **Stretches and Compressions**: These adjust the graph's scale.
- Vertical stretching is demonstrated by \( 4f(x) = 8 - 4x^2 \), scaling the graph vertically by a factor of 4, making it steeper.

3. **Reflections**: Reflecting a graph over the y-axis can be seen in \( f(-x) \). In this exercise, the function \( f(x) = 2 - x^2 \) is symmetrical about the y-axis, leading to \( f(-x) = f(x) \).

Understanding these transformations helps visualize how a function alters under different operations, a useful skill in graphing and calculus.

Quadratic functions are a specific type of polynomial function represented in the standard form \( ax^2 + bx + c \), where \( a eq 0 \). The function \( f(x) = 2 - x^2 \) can be rewritten as \( -x^2 + 2 \), showing it's a quadratic function with a leading coefficient of \(-1\).

Some important characteristics of quadratic functions include:

• The graph of a quadratic function is a parabola. If \( a \) is positive, the parabola opens upwards, and if \( a \) is negative, it opens downwards.
• The vertex of the parabola is its highest or lowest point. For \( f(x) = 2 - x^2 \), the vertex is at \( (0, 2) \).
• The axis of symmetry is a vertical line that divides the parabola into two mirror images. For the function \( -x^2 + 2 \), the axis of symmetry is \( x = 0 \).

Understanding the structure of quadratic functions aids in solving equations and inequalities, as well as in graphing and analyzing real-world problems.