How to find the slope of a quadratic function?
Understand the Problem
The question is asking for the method to determine the slope of a quadratic function, which typically involves taking the derivative of the function to find the slope at a given point.
Answer
The slope of a quadratic function at $x = x_0$ is $f'(x_0) = 2a(x_0) + b$.
Answer for screen readers
The slope of the quadratic function at a point $x = x_0$ is given by the formula: $$ f'(x_0) = 2a(x_0) + b $$
Steps to Solve
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Identify the quadratic function Start with the general form of a quadratic function, which is given by: $$ f(x) = ax^2 + bx + c $$ Where $a$, $b$, and $c$ are constants.
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Differentiate the function To find the slope of the quadratic function, take the derivative with respect to $x$. The derivative of the function is given by: $$ f'(x) = \frac{d}{dx}(ax^2 + bx + c) $$ Using the power rule, the derivative will be: $$ f'(x) = 2ax + b $$
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Evaluate the derivative at a specific point If you want to determine the slope at a specific point, say $x = x_0$, substitute this value into the derivative: $$ f'(x_0) = 2a(x_0) + b $$
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Interpret the result The resulting value, $f'(x_0)$, gives you the slope of the quadratic function at the point $x = x_0$.
The slope of the quadratic function at a point $x = x_0$ is given by the formula: $$ f'(x_0) = 2a(x_0) + b $$
More Information
The slope obtained from the derivative represents the rate of change of the function at that particular point. In the context of a graph, this means how steep the graph is at the point $(x_0, f(x_0))$. The concept of derivatives is fundamental to calculus and is used in various applications such as physics and engineering.
Tips
- Forgetting to differentiate the constant term $c$ which becomes $0$ and doesn't appear in the derivative.
- Incorrect application of the power rule, especially for higher degree polynomials.
