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When dealing with a quadratic function, you're encountering one of the simplest types of polynomial functions. A quadratic function has the general form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These functions create a parabola when graphed on a coordinate plane.

The basic features of a parabola include its vertex, axis of symmetry, and direction of opening. The vertex is the highest or lowest point on the graph, depending on whether the function opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)). For the given function \( f(x) = 2 - 3x + x^2 \), it's a standard quadratic function. As you can see, this function creates a parabola that opens upwards since the coefficient of \( x^2 \) is positive.

Understanding these basic properties provides a solid foundation for further calculus concepts, such as differentiation.

The power rule is a fundamental concept in calculus, used for finding derivatives of polynomial functions easily. It states that the derivative of \( x^n \) is \( n \cdot x^{n-1} \). This rule simplifies the process by allowing you to systematically work through polynomial terms.

In our given example, \( f(x) = 2 - 3x + x^2 \), applying the power rule:

• The derivative of the constant \( 2 \) becomes \( 0 \) because constants disappear when differentiated.
• The derivative of \( -3x \) is \( -3 \), as you multiply by the exponent \( 1 \) and reduce it by one (\( x^{1-1} \)).
• The derivative of \( x^2 \) is \( 2x \), multiplying by the exponent \( 2 \) and reducing it to \( x^{2-1} \).

So, the power rule guides us to find the derivative \( f'(x) = 2x - 3 \). This step is crucial in understanding how individual function components contribute to the behavior of the entire function.

Evaluating the derivative at a specific point allows us to understand the function's behavior in a particular spot. It's like zooming in on the function to see how it changes at that instant.

For the function \( f(x) = 2 - 3x + x^2 \), after differentiating, you get \( f'(x) = 2x - 3 \).
You are tasked with evaluating this derivative at \( x = -1 \).

To do this, plug \( -1 \) into \( f'(x) \):

• First, substitute \( -1 \) into the derivative: \( f'(-1) = 2(-1) - 3 \).
• Next, calculate the result: \(-2 - 3 = -5 \).

This calculation shows that the rate of change, or slope, of the function at \( x = -1 \) is \( -5 \). This means that at \( x = -1 \), the function is decreasing steeply. Evaluating derivatives helps in identifying not just slopes but also finding critical points and analyzing the behavior of a function in different contexts.