At what point does the function f(x) = 4 - x^2 have a constant rate of change? A) At the vertex B) x = 1 C) x = 0 D) x = -2
Understand the Problem
The question is asking us to determine the point at which the function f(x) = 4 - x^2 has a constant rate of change. This relates to understanding the properties of the function and its derivative, as the rate of change of a function is constant when it is a linear function. Since f(x) is a quadratic equation, we need to find out if it has any specific points where the rate of change could be constant.
Answer
The point is $(0, 4)$.
Answer for screen readers
The point at which the function has a constant rate of change is $(0, 4)$.
Steps to Solve
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Understand the Function's Nature Determine the type of function we are working with. The function given is $f(x) = 4 - x^2$, which is a quadratic function.
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Find the Derivative To find the rate of change, we compute the derivative of the function. The derivative will give us the rate of change of the function at any point $x$. $$ f'(x) = \frac{d}{dx}(4 - x^2) = -2x $$
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Set the Derivative to Zero We set the derivative equal to zero to find any points where the rate of change is constant (in this case, constant means it does not change, or is zero). $$ -2x = 0 $$
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Solve for x Now, we solve the equation obtained from the previous step. $$ x = 0 $$
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Find the Corresponding f(x) Value Finally, we substitute $x = 0$ back into the original function to find the corresponding $f(x)$ value. $$ f(0) = 4 - (0)^2 = 4 $$
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State the Point Thus, the point at which the function has a constant rate of change is at the coordinates $(0, 4)$.
The point at which the function has a constant rate of change is $(0, 4)$.
More Information
In the context of calculus, a constant rate of change occurs where the derivative is zero. For this quadratic function, the point $(0, 4)$ corresponds to the vertex, where the function changes direction, meaning the slope (rate of change) is horizontal at that point.
Tips
- A common mistake is to think that quadratic functions have constant rates of change everywhere. They only have a constant rate of change at specific points, often where the derivative equals zero.
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