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Prove that the Linear Function $f(x)=3x-7$ Changes by Equal Amounts Over Equal Intervals of Length 5

Step 1: Choose Two Input Values for $x$

To prove that the linear function $f(x)=3x-7$ changes by equal amounts over equal intervals of length 5, we need to start by choosing two input values for $x$ . Let's use $s$ and $s+5$ . We use two values of $x$ where the difference between them is 5, which is the length of the interval we are interested in.

Step 2: Calculate the Corresponding Output Values

Now that we have chosen our input values, we need to calculate the corresponding output values. We can do this by plugging $s$ and $s+5$ into the function $f(x)=3x-7$ .

For $x=s$ , we have:

$f(s) = 3s - 7$

For $x=s+5$ , we have:

$f(s+5) = 3(s+5) - 7$

Expanding the expression for $f(s+5)$ , we get:

$f(s+5) = 3s + 15 - 7$

Simplifying the expression, we get:

$f(s+5) = 3s + 8$

Step 3: Calculate the Change in the Function Over the Interval

Now that we have calculated the output values for $x=s$ and $x=s+5$ , we can calculate the change in the function over the interval. We can do this by subtracting the output value for $x=s$ from the output value for $x=s+5$ .

The change in the function over the interval is:

$\Delta f = f(s+5) - f(s)$

Substituting the expressions we calculated earlier, we get:

$\Delta f = (3s + 8) - (3s - 7)$

Simplifying the expression, we get:

$\Delta f = 15$

Step 4: Analyze the Result

Now that we have calculated the change in the function over the interval, we can analyze the result. We can see that the change in the function over the interval is 15, which is a constant value. This means that the linear function $f(x)=3x-7$ changes by equal amounts over equal intervals of length 5.

Conclusion

In this article, we proved that the linear function $f(x)=3x-7$ changes by equal amounts over equal intervals of length 5. We started by choosing two input values for $x$ , $s$ and $s+5$ , and then calculated the corresponding output values. We then calculated the change in the function over the interval and found that it was a constant value of 15. This means that the linear function $f(x)=3x-7$ is a linear function, and it changes by equal amounts over equal intervals of length 5.

The Importance of Linear Functions

Linear functions are an important concept in mathematics, and they have many real-world applications. They are used to model linear relationships between variables, and they can be used to make predictions and forecasts. Linear functions are also used in many fields, such as economics, physics, and engineering.

The Characteristics of Linear Functions

Linear functions have several characteristics that make them useful for modeling and predicting real-world phenomena. Some of the key characteristics of linear functions include:

Linearity: Linear functions are linear, meaning that they can be represented by a straight line.
Constant Slope: Linear functions have a constant slope, which means that the rate of change of the function is constant.
Constant Rate of Change: Linear functions have a constant rate of change, which means that the function changes by equal amounts over equal intervals.

The Applications of Linear Functions

Linear functions have many real-world applications, including:

Economics: Linear functions are used to model the relationship between variables such as price and quantity demanded.
Physics: Linear functions are used to model the motion of objects, such as the trajectory of a projectile.
Engineering: Linear functions are used to model the behavior of systems, such as the flow of fluids through pipes.

Conclusion

In conclusion, linear functions are an important concept in mathematics, and they have many real-world applications. They are used to model linear relationships between variables, and they can be used to make predictions and forecasts. Linear functions are also used in many fields, such as economics, physics, and engineering.
Q&A: Linear Functions and Their Applications

Q: What is a linear function?

A: A linear function is a function that can be represented by a straight line. It is a function that has a constant slope and a constant rate of change.

Q: What are the characteristics of a linear function?

A: The characteristics of a linear function include:

Linearity: Linear functions are linear, meaning that they can be represented by a straight line.
Constant Slope: Linear functions have a constant slope, which means that the rate of change of the function is constant.
Constant Rate of Change: Linear functions have a constant rate of change, which means that the function changes by equal amounts over equal intervals.

Q: What are some examples of linear functions?

A: Some examples of linear functions include:

f(x) = 2x + 3: This is a linear function that represents a straight line with a slope of 2 and a y-intercept of 3.
f(x) = x - 4: This is a linear function that represents a straight line with a slope of 1 and a y-intercept of -4.
f(x) = 3x - 2: This is a linear function that represents a straight line with a slope of 3 and a y-intercept of -2.

Q: What are some real-world applications of linear functions?

A: Some real-world applications of linear functions include:

Economics: Linear functions are used to model the relationship between variables such as price and quantity demanded.
Physics: Linear functions are used to model the motion of objects, such as the trajectory of a projectile.
Engineering: Linear functions are used to model the behavior of systems, such as the flow of fluids through pipes.

Q: How do I determine if a function is linear or not?

A: To determine if a function is linear or not, you can use the following criteria:

Check if the function is a straight line: If the function is a straight line, it is likely to be a linear function.
Check if the function has a constant slope: If the function has a constant slope, it is likely to be a linear function.
Check if the function has a constant rate of change: If the function has a constant rate of change, it is likely to be a linear function.

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:

Not checking if the function is a straight line: Make sure to check if the function is a straight line before assuming it is a linear function.
Not checking if the function has a constant slope: Make sure to check if the function has a constant slope before assuming it is a linear function.
Not checking if the function has a constant rate of change: Make sure to check if the function has a constant rate of change before assuming it is a linear function.

Q: How do I graph a linear function?

A: To graph a linear function, you can use the following steps:

Plot two points on the graph: Choose two points on the graph that satisfy the equation of the function.
Draw a straight line through the points: Draw a straight line through the two points to create the graph of the function.
Label the axes: Label the x-axis and y-axis to create a coordinate system.

Q: What are some common applications of linear functions in real-world scenarios?

A: Some common applications of linear functions in real-world scenarios include:

Modeling population growth: Linear functions can be used to model population growth and predict future population sizes.
Modeling economic trends: Linear functions can be used to model economic trends and predict future economic outcomes.
Modeling physical systems: Linear functions can be used to model physical systems, such as the motion of objects and the flow of fluids through pipes.

Conclusion

In conclusion, linear functions are an important concept in mathematics, and they have many real-world applications. They are used to model linear relationships between variables, and they can be used to make predictions and forecasts. By understanding the characteristics of linear functions and how to graph them, you can apply them to a wide range of real-world scenarios.