- A direct variation function follows the form \(f(x) = kx\).
- Using the given data \(f(3) = -9\), calculate the constant \(k = \frac{-9}{3} = -3\).
- Substitute the value of \(k\) back into the formula to get the final equation.
- The correct direct variation equation is \(\boxed{f(x)=-3x}\).
Explanation
- Understanding Direct Variation
The problem states that \(f(x)\) varies directly with \(x\). This is a foundational concept in algebra, describing a proportional relationship where the ratio of the output \(f(x)\) to the input \(x\) is always constant. This relationship can be expressed by the equation \(f(x) = kx\), where \(k\) is known as the constant of variation or the constant of proportionality. We are given the condition that \(f(3) = -9\). Our goal is to determine the specific constant \(k\) and establish the final equation for \(f(x)\).
- Determining the Constant of Variation and the Final Equation
To find the constant of variation \(k\), we substitute the given values \(x = 3\) and \(f(x) = -9\) into the formula \(f(x) = kx\). This gives us:
\[-9 = k \cdot 3\]
Now, we solve for \(k\) by dividing both sides of the equation by 3:
\[k = \frac{-9}{3} = -3\]
With the value of \(k\) determined, we can substitute it back into the general form \(f(x) = kx\) to obtain the specific direct variation equation:
\[f(x) = -3x\]
- Conclusion and Final Answer
Comparing our derived equation \(f(x) = -3x\) with the given choices:
- \(f(x) = -3x\)
- \(f(x) = -12x\)
- \(f(x) = -\frac{1}{3}x\)
- \(f(x) = -6x\)
Our result matches the first option. Therefore, the direct variation equation is \(f(x) = -3x\). The final answer is \(\boxed{f(x)=-3x}\).
Examples
Direct variation is very useful in everyday life, especially when dealing with constant rates. For example, if you are driving at a constant speed, the distance you travel, \(d\), varies directly with the time, \(t\), spent driving. If you travel 150 miles in 3 hours, your constant of variation is the speed, \(k = \frac{150}{3} = 50\), so your distance formula is \(d = 50t\). This linear relationship allows for easy calculation of distance at any given time, just like how we identified the function \(f(x) = -3x\) in our problem.