The Effect of Translation on the Function

Oct 13, 2025 by miodeya.com 104 views

Understanding the Original Function

The original function given is ${$ f(x)=x^2$}$. This is a quadratic function that represents a parabola opening upwards with its vertex at the origin (0, 0). The x-intercepts of this function can be found by setting ${$ f(x)=0$}$ and solving for x.

The Effect of Translation on the Function

When the function is translated down 4 units, the new function becomes ${$ f(x)=(x^2)-4$}$. This means that the entire parabola is shifted downwards by 4 units, resulting in a new vertex at (0, -4).

Determining the x-Intercept(s) of the Translated Function

To find the x-intercepts of the translated function, we need to set ${$ f(x)=0$}$ and solve for x. This gives us the equation ${$ (x^2)-4=0$}$.

Solving the Equation

We can solve this equation by adding 4 to both sides, resulting in ${$ x^2=4$}$. Taking the square root of both sides gives us ${$ x=\pm\sqrt{4}$}$, which simplifies to ${$ x=\pm2$}$.

Conclusion

Based on the solution to the equation, we can conclude that the x-intercept(s) of the translated function are ${$ x=-2$}$ and ${$ x=2$}$. Therefore, the correct answer is:

C. ${$ x=-2$}$

Why the Other Options are Incorrect

Option A is incorrect because the x-intercepts of the translated function are not ${$ x=-16$}$ and ${$ x=16$}$. Option B is incorrect because the translated function does have x-intercepts, which we have just found to be ${$ x=-2$}$ and ${$ x=2$}$.

Key Takeaways

When a function is translated down, the x-intercepts of the original function are shifted downwards by the same amount.
To find the x-intercepts of a translated function, we need to set the function equal to 0 and solve for x.
The x-intercepts of a quadratic function can be found by taking the square root of the constant term in the equation.

Real-World Applications

Understanding how to find the x-intercepts of a translated function is important in a variety of real-world applications, such as:

Physics: When modeling the motion of an object, we may need to translate the position function down to account for a change in the object's position.
Engineering: When designing a system, we may need to translate the function that represents the system's behavior down to account for a change in the system's parameters.
Economics: When modeling the behavior of a market, we may need to translate the function that represents the market's equilibrium down to account for a change in the market's conditions.

Conclusion

Frequently Asked Questions

Q: What is the effect of translating a function down on its x-intercepts?

A: When a function is translated down, the x-intercepts of the original function are shifted downwards by the same amount.

Q: How do I find the x-intercepts of a translated function?

A: To find the x-intercepts of a translated function, we need to set the function equal to 0 and solve for x.

Q: What is the equation for finding the x-intercepts of a translated function?

A: The equation for finding the x-intercepts of a translated function is ${$ f(x)=0$}$.

Q: How do I solve the equation ${$ f(x)=0$}$?

A: To solve the equation ${$ f(x)=0$}$, we need to add 4 to both sides, resulting in ${$ x^2=4$}$. Taking the square root of both sides gives us ${$ x=\pm\sqrt{4}$}$, which simplifies to ${$ x=\pm2$}$.

Q: What are the x-intercepts of the translated function ${$ f(x)=(x^2)-4$}$?

A: The x-intercepts of the translated function ${$ f(x)=(x^2)-4$}$ are ${$ x=-2$}$ and ${$ x=2$}$.

Q: Why are the x-intercepts of the translated function not ${$ x=-16$}$ and ${$ x=16$}$?

A: The x-intercepts of the translated function are not ${$ x=-16$}$ and ${$ x=16$}$ because the translated function is a quadratic function that opens upwards, and its x-intercepts are determined by the equation ${$ x^2=4$}$, which has solutions ${$ x=\pm2$}$.

Q: What are some real-world applications of determining the x-intercept(s) of a translated function?

A: Some real-world applications of determining the x-intercept(s) of a translated function include:

Physics: When modeling the motion of an object, we may need to translate the position function down to account for a change in the object's position.
Engineering: When designing a system, we may need to translate the function that represents the system's behavior down to account for a change in the system's parameters.
Economics: When modeling the behavior of a market, we may need to translate the function that represents the market's equilibrium down to account for a change in the market's conditions.

Q: How can I apply the knowledge of determining the x-intercept(s) of a translated function to my own work?

A: You can apply the knowledge of determining the x-intercept(s) of a translated function to your own work by:

Using the equation ${$ f(x)=0$}$ to find the x-intercepts of a translated function.
Understanding how translation affects the function's behavior.
Applying the knowledge to real-world applications, such as physics, engineering, and economics.

Q: What are some common mistakes to avoid when determining the x-intercept(s) of a translated function?

A: Some common mistakes to avoid when determining the x-intercept(s) of a translated function include:

Not setting the function equal to 0.
Not solving for x.
Not understanding how translation affects the function's behavior.

Q: How can I practice determining the x-intercept(s) of a translated function?

A: You can practice determining the x-intercept(s) of a translated function by:

Solving equations of the form ${$ f(x)=0$}$.
Graphing the translated function and identifying its x-intercepts.
Applying the knowledge to real-world applications.

Conclusion

In conclusion, determining the x-intercept(s) of a translated function requires a clear understanding of how translation affects the function's behavior. By following the steps outlined in this article and practicing with examples, you can develop the skills and knowledge needed to apply this concept to a variety of real-world applications.