Write A Function Rule For The Output Is Half Of The Input $x$.$y = \frac{1}{2}x$
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Introduction
In mathematics, a function rule is a mathematical expression that describes the relationship between the input and output of a function. In this article, we will discuss the function rule for "The output is half of the input $x$." The function rule is given by $y = \frac{1}{2}x$.
Understanding the Function Rule
The function rule $y = \frac{1}{2}x$ states that the output $y$ is equal to half of the input $x$. This means that if we input a value of $x$, the output $y$ will be half of that value.
Example 1: Input Value of 4
Let's say we input a value of $x = 4$. Using the function rule $y = \frac{1}{2}x$, we can calculate the output $y$ as follows:
So, if we input a value of $x = 4$, the output $y$ will be $2$.
Example 2: Input Value of 10
Let's say we input a value of $x = 10$. Using the function rule $y = \frac{1}{2}x$, we can calculate the output $y$ as follows:
So, if we input a value of $x = 10$, the output $y$ will be $5$.
Graphing the Function
To visualize the function rule $y = \frac{1}{2}x$, we can graph it on a coordinate plane. The graph will be a straight line with a slope of $\frac{1}{2}$ and a y-intercept of $0$.
Graphing the Function: Step-by-Step
- Step 1: Draw a coordinate plane with x-axis and y-axis.
- Step 2: Choose a value of $x$, say $x = 4$.
- Step 3: Calculate the corresponding value of $y$ using the function rule $y = \frac{1}{2}x$.
- Step 4: Plot the point $(x, y)$ on the coordinate plane.
- Step 5: Repeat steps 2-4 for different values of $x$ to get multiple points on the graph.
- Step 6: Connect the points to form a straight line.
Real-World Applications
The function rule $y = \frac{1}{2}x$ has many real-world applications. For example:
- Discounts: If a store offers a 50% discount on all items, the price of each item will be half of the original price. This can be represented by the function rule $y = \frac{1}{2}x$.
- Interest Rates: If a bank offers a 50% interest rate on a savings account, the interest earned will be half of the principal amount. This can be represented by the function rule $y = \frac{1}{2}x$.
- Scaling: If we want to scale down an object by half, the new size will be half of the original size. This can be represented by the function rule $y = \frac{1}{2}x$.
Conclusion
In conclusion, the function rule $y = \frac{1}{2}x$ represents the relationship between the input and output of a function where the output is half of the input. This function rule has many real-world applications and can be graphed on a coordinate plane. By understanding the function rule and its applications, we can better analyze and solve problems in mathematics and real-world scenarios.
References
- Mathematics Textbook: "Functions and Graphs" by [Author's Name]
- Online Resource: "Function Rules" by [Website Name]
Further Reading
- Linear Functions: "Understanding Linear Functions" by [Author's Name]
- Graphing Functions: "Graphing Functions on a Coordinate Plane" by [Author's Name]
Glossary
- Function Rule: A mathematical expression that describes the relationship between the input and output of a function.
- Input: The value that is input into a function.
- Output: The value that is output from a function.
- Graph: A visual representation of a function on a coordinate plane.
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Introduction
In our previous article, we discussed the function rule $y = \frac{1}{2}x$, which represents the relationship between the input and output of a function where the output is half of the input. In this article, we will answer some frequently asked questions about the function rule and its applications.
Q&A
Q: What is the function rule for half of the input?
A: The function rule for half of the input is $y = \frac{1}{2}x$.
Q: How do I calculate the output using the function rule?
A: To calculate the output using the function rule, simply multiply the input value by $\frac{1}{2}$.
Q: What is the y-intercept of the function rule?
A: The y-intercept of the function rule is $0$.
Q: How do I graph the function rule on a coordinate plane?
A: To graph the function rule on a coordinate plane, choose a value of $x$, calculate the corresponding value of $y$ using the function rule, and plot the point $(x, y)$ on the coordinate plane. Repeat this process for different values of $x$ to get multiple points on the graph, and then connect the points to form a straight line.
Q: What are some real-world applications of the function rule?
A: Some real-world applications of the function rule include:
- Discounts: If a store offers a 50% discount on all items, the price of each item will be half of the original price.
- Interest Rates: If a bank offers a 50% interest rate on a savings account, the interest earned will be half of the principal amount.
- Scaling: If we want to scale down an object by half, the new size will be half of the original size.
Q: Can I use the function rule to solve problems in mathematics and real-world scenarios?
A: Yes, you can use the function rule to solve problems in mathematics and real-world scenarios. By understanding the function rule and its applications, you can better analyze and solve problems.
Q: How do I determine if a function is linear or non-linear?
A: To determine if a function is linear or non-linear, look at the graph of the function. If the graph is a straight line, the function is linear. If the graph is a curve, the function is non-linear.
Q: Can I use the function rule to graph other functions?
A: Yes, you can use the function rule to graph other functions. By modifying the function rule, you can graph different types of functions.
Conclusion
In conclusion, the function rule $y = \frac{1}{2}x$ represents the relationship between the input and output of a function where the output is half of the input. This function rule has many real-world applications and can be graphed on a coordinate plane. By understanding the function rule and its applications, you can better analyze and solve problems in mathematics and real-world scenarios.
References
- Mathematics Textbook: "Functions and Graphs" by [Author's Name]
- Online Resource: "Function Rules" by [Website Name]
Further Reading
- Linear Functions: "Understanding Linear Functions" by [Author's Name]
- Graphing Functions: "Graphing Functions on a Coordinate Plane" by [Author's Name]
Glossary
- Function Rule: A mathematical expression that describes the relationship between the input and output of a function.
- Input: The value that is input into a function.
- Output: The value that is output from a function.
- Graph: A visual representation of a function on a coordinate plane.
- Linear Function: A function that can be represented by a straight line on a coordinate plane.
- Non-Linear Function: A function that cannot be represented by a straight line on a coordinate plane.