Use the techniques of this chapter to find the vertex and...

Use the techniques of this chapter to find the vertex and intervals where $f$ is increasing and decreasing, given
$$f(x)=a x^{2}+b x+c$$
where we assume $a>0 .$ Verify that this agrees with what we found in Chapter 10 .

Key Concepts

Quadratic Functions

A quadratic function is a polynomial of degree two which has the general form f(x) = ax² + bx + c. This type of function is characterized by a parabolic graph that opens upward when a > 0 or downward when a < 0. Quadratic functions are fundamental in algebra and calculus, serving as one of the simplest examples of polynomial behavior with a clear turning point known as the vertex.

Vertex of a Parabola

The vertex of a parabola is the point where the quadratic function attains its minimum or maximum value. It can be found using the formula (-b/(2a), f(-b/(2a))). The vertex is essential in understanding the graph's shape, as it represents the transition point between increasing and decreasing behavior. For a quadratic function with a > 0, the vertex corresponds to the minimum point on the graph.

Derivatives

In calculus, the derivative of a function represents its rate of change. For a quadratic function f(x) = ax² + bx + c, the derivative is f'(x) = 2ax + b. This derivative gives insight into the slope of the tangent line at any point on the curve and is a key tool in determining where the function is increasing or decreasing by analyzing the sign of the derivative.

Increasing and Decreasing Intervals

The increasing and decreasing intervals of a function are determined by the sign of its derivative. If the derivative f'(x) is positive over an interval, the function is increasing on that interval; if f'(x) is negative, the function is decreasing. For the quadratic function f(x) = ax² + bx + c with a > 0, there is a critical point at x = -b/(2a) which marks the transition from decreasing (for x < -b/(2a)) to increasing (for x > -b/(2a)).

Transcript

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