Select All That Apply.The Slope Of The Graph Of A Direct Variation Function Is -3. Which Of The Following Is True?- The Function Is Linear.- The Equation Is Y = − 3 X Y = -3x Y = − 3 X .- The Equation Of The Function Is Y = − 1 3 X Y = -\frac{1}{3}x Y = − 3 1 ​ X .- The Point

Direct variation functions are a type of linear function where the output value is directly proportional to the input value. In other words, as one variable increases, the other variable also increases at a constant rate. The slope of the graph of a direct variation function represents the constant rate of change between the variables.

The Slope of a Direct Variation Function

The slope of a direct variation function is a key characteristic that distinguishes it from other types of functions. In this case, we are given that the slope of the graph of a direct variation function is -3. This means that for every unit increase in the input variable, the output variable decreases by 3 units.

Analyzing the Options

Now, let's analyze the options given to determine which one is true.

Option 1: The function is linear

A direct variation function is indeed a type of linear function. The graph of a direct variation function is a straight line, and the slope of the line represents the constant rate of change between the variables. Therefore, option 1 is true.

Option 2: The equation is $y = -3x$

The equation $y = -3x$ represents a direct variation function with a slope of -3. However, this equation is not the only possible equation for a direct variation function with a slope of -3. There are other equations that could represent the same function, such as $y = -3x + 2$ or $y = -3x - 5$. Therefore, option 2 is not necessarily true.

Option 3: The equation of the function is $y = -\frac{1}{3}x$

This option is incorrect because the slope of the function is -3, not -\frac{1}{3}. The equation $y = -\frac{1}{3}x$ represents a direct variation function with a slope of -\frac{1}{3}, not -3.

Option 4: The point (3, -9) is on the graph

To determine if the point (3, -9) is on the graph, we need to substitute x = 3 into the equation of the function. Let's assume the equation is $y = -3x$. Substituting x = 3, we get y = -3(3) = -9. Therefore, the point (3, -9) is indeed on the graph.

Conclusion

In conclusion, the correct answer is option 1: The function is linear. The slope of the graph of a direct variation function is -3, and this function is indeed a type of linear function.

Direct Variation Functions: Key Concepts

• Definition: A direct variation function is a type of linear function where the output value is directly proportional to the input value.
• Slope: The slope of a direct variation function represents the constant rate of change between the variables.
• Equation: The equation of a direct variation function can be written in the form $y = mx$, where m is the slope and x is the input variable.
• Graph: The graph of a direct variation function is a straight line with a constant slope.

Examples of Direct Variation Functions

• Example 1: The cost of a product is directly proportional to the number of units sold. If the cost is $10 per unit, the equation of the function is $y = 10x$, where x is the number of units sold and y is the total cost.
• Example 2: The distance traveled by a car is directly proportional to the time traveled. If the car travels 60 miles per hour, the equation of the function is $y = 60x$, where x is the time traveled in hours and y is the distance traveled in miles.

Real-World Applications of Direct Variation Functions

Direct variation functions have many real-world applications, including:

• Business: Direct variation functions can be used to model the cost of a product, the revenue generated by a business, or the number of units sold.
• Science: Direct variation functions can be used to model the distance traveled by an object, the force applied to an object, or the energy transferred to an object.
• Engineering: Direct variation functions can be used to model the stress applied to a material, the strain on a material, or the energy transferred to a system.

Conclusion

Q: What is a direct variation function?

A: A direct variation function is a type of linear function where the output value is directly proportional to the input value. In other words, as one variable increases, the other variable also increases at a constant rate.

Q: What is the slope of a direct variation function?

A: The slope of a direct variation function represents the constant rate of change between the variables. It is a key characteristic that distinguishes direct variation functions from other types of functions.

Q: How do I determine the equation of a direct variation function?

A: To determine the equation of a direct variation function, you need to know the slope and one point on the graph. You can use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point on the graph.

Q: What is the difference between a direct variation function and a linear function?

A: A direct variation function is a type of linear function where the output value is directly proportional to the input value. A linear function, on the other hand, is a function that can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Q: Can a direct variation function have a negative slope?

A: Yes, a direct variation function can have a negative slope. This means that as the input variable increases, the output variable decreases at a constant rate.

Q: Can a direct variation function have a zero slope?

A: No, a direct variation function cannot have a zero slope. This is because the slope of a direct variation function represents the constant rate of change between the variables, and a zero slope would mean that there is no change.

Q: Can a direct variation function have a fractional slope?

A: Yes, a direct variation function can have a fractional slope. This means that the output variable increases or decreases at a rate that is a fraction of the input variable.

Q: How do I graph a direct variation function?

A: To graph a direct variation function, you can use the slope and one point on the graph to determine the equation of the function. You can then plot the graph using a coordinate plane.

Q: Can a direct variation function be used to model real-world situations?

A: Yes, direct variation functions can be used to model real-world situations, such as the cost of a product, the revenue generated by a business, or the number of units sold.

Q: What are some examples of direct variation functions in real-world situations?

A: Some examples of direct variation functions in real-world situations include:

• The cost of a product is directly proportional to the number of units sold.
• The distance traveled by a car is directly proportional to the time traveled.
• The force applied to an object is directly proportional to the distance over which the force is applied.

Q: How do I determine the equation of a direct variation function from a graph?

A: To determine the equation of a direct variation function from a graph, you need to identify the slope and one point on the graph. You can then use the point-slope form of a linear equation to determine the equation of the function.

Q: Can a direct variation function be used to model a situation where the output variable decreases as the input variable increases?

A: Yes, a direct variation function can be used to model a situation where the output variable decreases as the input variable increases. This is known as a negative direct variation function.

Q: How do I determine the equation of a negative direct variation function?

A: To determine the equation of a negative direct variation function, you need to know the slope and one point on the graph. You can then use the point-slope form of a linear equation to determine the equation of the function.

Q: Can a direct variation function be used to model a situation where the output variable increases at a rate that is not constant?

A: No, a direct variation function cannot be used to model a situation where the output variable increases at a rate that is not constant. This is because the slope of a direct variation function represents the constant rate of change between the variables.

Q: How do I determine the equation of a direct variation function from a set of data points?

A: To determine the equation of a direct variation function from a set of data points, you need to identify the slope and one point on the graph. You can then use the point-slope form of a linear equation to determine the equation of the function.

Q: Can a direct variation function be used to model a situation where the output variable is not directly proportional to the input variable?

A: No, a direct variation function cannot be used to model a situation where the output variable is not directly proportional to the input variable. This is because the slope of a direct variation function represents the constant rate of change between the variables.

Conclusion

In conclusion, direct variation functions are a type of linear function where the output value is directly proportional to the input value. The slope of a direct variation function represents the constant rate of change between the variables. Direct variation functions have many real-world applications, including business, science, and engineering.