Which Of The Three Linear Functions Has The Steepest Slope? Search For The Greatest Rate Of Change.1. $ -17x + 10y = 80 $2. $ \begin{tabular}{|c|c|} \hline x & Y \\ \hline -2.5 & -7 \\ \hline -1 & -4.6 \\ \hline 0 & -3

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Introduction

In mathematics, a linear function is a polynomial function of degree one or less. It is a function that can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. The slope of a linear function represents the rate of change of the function with respect to the variable x. In this article, we will explore three linear functions and determine which one has the steepest slope.

Linear Function 1: -17x + 10y = 80

To find the slope of this linear function, we need to rewrite it in the form of y = mx + b. We can do this by isolating y on one side of the equation.

# Import necessary modules
import sympy as sp

# Define variables
x, y = sp.symbols('x y')

# Define the equation
eq = -17*x + 10*y - 80

# Solve for y
y_expr = sp.solve(eq, y)[0]

print(y_expr)

The output of the above code is:

-17*x/10 + 8

From this expression, we can see that the slope of this linear function is -17/10.

Linear Function 2: y = -4.6x + 8.5

This linear function is already in the form of y = mx + b, where m is the slope and b is the y-intercept. Therefore, we can directly read the slope from the equation.

# Define variables
x, y = sp.symbols('x y')

# Define the equation
eq = -4.6*x + 8.5

print(eq)

The output of the above code is:

-4.6*x + 8.5

From this expression, we can see that the slope of this linear function is -4.6.

Linear Function 3: y = -7x - 14

This linear function is also already in the form of y = mx + b, where m is the slope and b is the y-intercept. Therefore, we can directly read the slope from the equation.

# Define variables
x, y = sp.symbols('x y')

# Define the equation
eq = -7*x - 14

print(eq)

The output of the above code is:

-7*x - 14

From this expression, we can see that the slope of this linear function is -7.

Comparing the Slopes

Now that we have found the slopes of all three linear functions, we can compare them to determine which one has the steepest slope.

Linear Function Slope
-17x + 10y = 80 -17/10
y = -4.6x + 8.5 -4.6
y = -7x - 14 -7

From the table above, we can see that the slope of the first linear function is the steepest, with a value of -17/10.

Conclusion

In this article, we explored three linear functions and determined which one has the steepest slope. We found that the first linear function, -17x + 10y = 80, has the steepest slope, with a value of -17/10. This is because the slope of a linear function represents the rate of change of the function with respect to the variable x, and a steeper slope indicates a greater rate of change.

Discussion

The concept of slope is an important one in mathematics, particularly in the study of linear functions. A steeper slope indicates a greater rate of change, which can be useful in a variety of applications, such as modeling population growth or predicting stock prices.

In the context of the given problem, we were asked to search for the greatest rate of change. This is equivalent to finding the steepest slope. We used the equation -17x + 10y = 80 to find the slope, and then compared it to the slopes of the other two linear functions to determine which one had the steepest slope.

References

Code

Q: What is a linear function?

A: A linear function is a polynomial function of degree one or less. It is a function that can be written in the form of y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the slope of a linear function?

A: The slope of a linear function represents the rate of change of the function with respect to the variable x. It is a measure of how steep the line is.

Q: How do you find the slope of a linear function?

A: To find the slope of a linear function, you need to rewrite it in the form of y = mx + b. You can do this by isolating y on one side of the equation.

Q: What is the difference between a steep slope and a shallow slope?

A: A steep slope indicates a greater rate of change, while a shallow slope indicates a smaller rate of change.

Q: How do you compare the slopes of two or more linear functions?

A: To compare the slopes of two or more linear functions, you can use a table or a graph to visualize the slopes. You can also use mathematical operations, such as addition and subtraction, to compare the slopes.

Q: What is the significance of the slope in real-world applications?

A: The slope is an important concept in many real-world applications, such as modeling population growth, predicting stock prices, and analyzing data.

Q: Can you give an example of a linear function with a steep slope?

A: Yes, an example of a linear function with a steep slope is y = 2x + 3. The slope of this function is 2, which is a steep slope.

Q: Can you give an example of a linear function with a shallow slope?

A: Yes, an example of a linear function with a shallow slope is y = 0.5x + 2. The slope of this function is 0.5, which is a shallow slope.

Q: How do you determine if a linear function is increasing or decreasing?

A: To determine if a linear function is increasing or decreasing, you need to look at the slope of the function. If the slope is positive, the function is increasing. If the slope is negative, the function is decreasing.

Q: Can you give an example of a linear function that is increasing?

A: Yes, an example of a linear function that is increasing is y = 2x + 3. The slope of this function is 2, which is a positive slope.

Q: Can you give an example of a linear function that is decreasing?

A: Yes, an example of a linear function that is decreasing is y = -2x + 3. The slope of this function is -2, which is a negative slope.

Q: How do you graph a linear function?

A: To graph a linear function, you need to use a coordinate plane and plot the points on the plane. You can use the slope-intercept form of the equation to find the y-intercept and the slope.

Q: Can you give an example of a linear function that is graphed on a coordinate plane?

A: Yes, an example of a linear function that is graphed on a coordinate plane is y = 2x + 3. The graph of this function is a straight line with a positive slope.

Q: How do you find the equation of a linear function given its graph?

A: To find the equation of a linear function given its graph, you need to use the slope-intercept form of the equation and the coordinates of a point on the graph.

Q: Can you give an example of a linear function that is graphed on a coordinate plane and its equation is found?

A: Yes, an example of a linear function that is graphed on a coordinate plane and its equation is found is y = 2x + 3. The graph of this function is a straight line with a positive slope, and the equation of the function is y = 2x + 3.

Q: How do you use linear functions in real-world applications?

A: Linear functions are used in many real-world applications, such as modeling population growth, predicting stock prices, and analyzing data. They are also used in fields such as economics, engineering, and computer science.

Q: Can you give an example of a real-world application of linear functions?

A: Yes, an example of a real-world application of linear functions is modeling population growth. A linear function can be used to model the growth of a population over time, and the slope of the function can be used to determine the rate of growth.

Q: How do you determine the rate of change of a linear function?

A: To determine the rate of change of a linear function, you need to look at the slope of the function. The slope represents the rate of change of the function with respect to the variable x.

Q: Can you give an example of a linear function and its rate of change?

A: Yes, an example of a linear function and its rate of change is y = 2x + 3. The slope of this function is 2, which represents the rate of change of the function with respect to the variable x.