True Or False: If $f(x)$ Is A Linear Function With A Graph That Extends Through Points At $(-2,4)$, \$(0,0)$[/tex\], And $(2,-4)$, Then It Has A Relative Minimum When $x=-2$.A. True B. False
Understanding Linear Functions
A linear function is a polynomial function of degree one, which means it can be written in the form of $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The graph of a linear function is a straight line, and it can be extended through any number of points.
Analyzing the Given Points
We are given three points through which the graph of the linear function $f(x)$ extends: $(-2,4)$, $(0,0)$, and $(2,-4)$. To determine the relative minimum of the function, we need to find the point on the graph where the function changes from decreasing to increasing.
Calculating the Slope
To find the slope of the linear function, we can use the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$. Using the points $(0,0)$ and $(2,-4)$, we get:
Determining the Equation of the Line
Now that we have the slope, we can use the point-slope form of a linear equation to find the equation of the line. Using the point $(0,0)$ and the slope $m = -2$, we get:
To find the value of $b$, we can substitute the point $(0,0)$ into the equation:
Finding the Relative Minimum
To find the relative minimum of the function, we need to find the point on the graph where the function changes from decreasing to increasing. This occurs when the slope of the function is zero. However, since the slope of the linear function is constant, it never changes from decreasing to increasing. Therefore, the function does not have a relative minimum.
Conclusion
Based on the analysis, we can conclude that the statement "If $f(x)$ is a linear function with a graph that extends through points at $(-2,4)$, $(0,0)$, and $(2,-4)$, then it has a relative minimum when $x=-2$" is False.
Key Takeaways
- A linear function is a polynomial function of degree one, which can be written in the form of $f(x) = mx + b$.
- The graph of a linear function is a straight line that can be extended through any number of points.
- The slope of a linear function is constant, and it never changes from decreasing to increasing.
- A relative minimum occurs when the function changes from decreasing to increasing, which is not possible for a linear function.
Real-World Applications
Linear functions have many real-world applications, including:
- Modeling population growth
- Describing the motion of objects
- Calculating the cost of goods
- Determining the demand for products
Common Mistakes
- Assuming that a linear function has a relative minimum
- Failing to recognize that the slope of a linear function is constant
- Not understanding the concept of relative minimum
Solutions to Common Mistakes
- Recognize that a linear function does not have a relative minimum.
- Understand that the slope of a linear function is constant.
- Review the concept of relative minimum and how it applies to different types of functions.
Understanding Linear Functions and Relative Minima
In our previous article, we discussed the concept of linear functions and relative minima. We analyzed a linear function with a graph that extended through points at $(-2,4)$, $(0,0)$, and $(2,-4)$, and concluded that it does not have a relative minimum.
Q&A: Linear Functions and Relative Minima
Q: What is a linear function?
A: A linear function is a polynomial function of degree one, which can be written in the form of $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
Q: What is a relative minimum?
A: A relative minimum is a point on the graph of a function where the function changes from decreasing to increasing.
Q: Does a linear function have a relative minimum?
A: No, a linear function does not have a relative minimum. The slope of a linear function is constant, and it never changes from decreasing to increasing.
Q: How can I determine if a function has a relative minimum?
A: To determine if a function has a relative minimum, you need to find the point on the graph where the function changes from decreasing to increasing. This can be done by finding the derivative of the function and setting it equal to zero.
Q: What are some real-world applications of linear functions?
A: Linear functions have many real-world applications, including:
- Modeling population growth
- Describing the motion of objects
- Calculating the cost of goods
- Determining the demand for products
Q: What are some common mistakes to avoid when working with linear functions?
A: Some common mistakes to avoid when working with linear functions include:
- Assuming that a linear function has a relative minimum
- Failing to recognize that the slope of a linear function is constant
- Not understanding the concept of relative minimum
Q: How can I review the concept of relative minimum and linear functions?
A: You can review the concept of relative minimum and linear functions by:
- Reading our previous article on the topic
- Watching video tutorials on the subject
- Practicing problems and exercises to reinforce your understanding
Additional Resources
- Khan Academy: Linear Functions
- Mathway: Linear Functions and Relative Minima
- Wolfram Alpha: Linear Functions and Relative Minima
Conclusion
We hope this Q&A guide has helped you understand the concept of linear functions and relative minima. Remember to avoid common mistakes and review the concept regularly to reinforce your understanding.
Key Takeaways
- A linear function is a polynomial function of degree one, which can be written in the form of $f(x) = mx + b$.
- The graph of a linear function is a straight line that can be extended through any number of points.
- The slope of a linear function is constant, and it never changes from decreasing to increasing.
- A relative minimum occurs when the function changes from decreasing to increasing, which is not possible for a linear function.
Real-World Applications
Linear functions have many real-world applications, including:
- Modeling population growth
- Describing the motion of objects
- Calculating the cost of goods
- Determining the demand for products
Common Mistakes
- Assuming that a linear function has a relative minimum
- Failing to recognize that the slope of a linear function is constant
- Not understanding the concept of relative minimum
Solutions to Common Mistakes
- Recognize that a linear function does not have a relative minimum.
- Understand that the slope of a linear function is constant.
- Review the concept of relative minimum and how it applies to different types of functions.