Timmy Writes The Equation F ( X ) = 1 4 X − 1 F(x)=\frac{1}{4} X-1 F ( X ) = 4 1 ​ X − 1 . He Then Doubles Both Of The Terms On The Right Side To Create The Equation G ( X ) = 1 2 X − 2 G(x)=\frac{1}{2} X-2 G ( X ) = 2 1 ​ X − 2 . How Does The Graph Of G ( X G(x G ( X ] Compare To The Graph Of F ( X F(x F ( X ]?A. The

Introduction

In mathematics, understanding the relationship between two linear equations and their corresponding graphs is crucial for solving various problems. In this article, we will explore how the graph of one linear equation compares to the graph of another linear equation that is derived from it by doubling both terms on the right side.

The Original Equation

The original equation is given by $f(x)=\frac{1}{4} x-1$. This is a linear equation in the form of $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope of the line is $\frac{1}{4}$, and the y-intercept is $-1$.

The Modified Equation

The modified equation is obtained by doubling both terms on the right side of the original equation. This results in the equation $g(x)=\frac{1}{2} x-2$. Again, this is a linear equation in the form of $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope of the line is $\frac{1}{2}$, and the y-intercept is $-2$.

Comparing the Graphs

To compare the graphs of the two equations, we need to understand how the changes in the slope and y-intercept affect the graph. When we double both terms on the right side of the original equation, the slope is multiplied by $2$, and the y-intercept is multiplied by $2$.

Effect of Doubling the Slope

When the slope is doubled, the graph of the line becomes steeper. This means that for a given change in $x$, the change in $y$ is greater. In other words, the line becomes more vertical.

Effect of Doubling the Y-Intercept

When the y-intercept is doubled, the graph of the line is shifted upward. This means that the line is now higher than the original line by the amount of the change in the y-intercept.

Visualizing the Graphs

To visualize the graphs of the two equations, we can use a graphing tool or software. By plotting the two equations on the same coordinate plane, we can see how the graph of $g(x)$ compares to the graph of $f(x)$.

Graph of $f(x)$

The graph of $f(x)$ is a line with a slope of $\frac{1}{4}$ and a y-intercept of $-1$. The line passes through the point $(0, -1)$ and has a positive slope.

Graph of $g(x)$

The graph of $g(x)$ is a line with a slope of $\frac{1}{2}$ and a y-intercept of $-2$. The line passes through the point $(0, -2)$ and has a positive slope.

Comparison of the Graphs

By comparing the two graphs, we can see that the graph of $g(x)$ is steeper than the graph of $f(x)$. This is because the slope of $g(x)$ is twice the slope of $f(x)$. Additionally, the graph of $g(x)$ is higher than the graph of $f(x)$ by the amount of the change in the y-intercept.

Conclusion

In conclusion, when we double both terms on the right side of a linear equation, the graph of the resulting equation is steeper and higher than the original graph. This is because the slope is multiplied by $2$, and the y-intercept is multiplied by $2$. By understanding how the changes in the slope and y-intercept affect the graph, we can compare the graphs of two linear equations and solve various problems in mathematics.

Key Takeaways

• Doubling both terms on the right side of a linear equation results in a steeper and higher graph.
• The slope is multiplied by $2$, and the y-intercept is multiplied by $2$.
• The graph of the resulting equation is compared to the original graph by plotting both equations on the same coordinate plane.

Further Exploration

• Explore how other changes to the slope and y-intercept affect the graph of a linear equation.
• Use graphing software or tools to visualize the graphs of different linear equations.
• Solve problems that involve comparing the graphs of two linear equations.
  Frequently Asked Questions (FAQs) =====================================

Q: What happens when we double both terms on the right side of a linear equation?

A: When we double both terms on the right side of a linear equation, the slope is multiplied by 2, and the y-intercept is multiplied by 2. This results in a steeper and higher graph.

Q: How does the graph of g(x) compare to the graph of f(x)?

A: The graph of g(x) is steeper than the graph of f(x) because the slope of g(x) is twice the slope of f(x). Additionally, the graph of g(x) is higher than the graph of f(x) by the amount of the change in the y-intercept.

Q: What is the effect of doubling the slope on the graph of a linear equation?

A: When the slope is doubled, the graph of the line becomes steeper. This means that for a given change in x, the change in y is greater. In other words, the line becomes more vertical.

Q: What is the effect of doubling the y-intercept on the graph of a linear equation?

A: When the y-intercept is doubled, the graph of the line is shifted upward. This means that the line is now higher than the original line by the amount of the change in the y-intercept.

Q: How can we visualize the graphs of two linear equations?

A: We can use a graphing tool or software to plot the two equations on the same coordinate plane. This allows us to see how the graph of g(x) compares to the graph of f(x).

Q: What are some real-world applications of comparing the graphs of two linear equations?

A: Comparing the graphs of two linear equations has many real-world applications, such as:

• Modeling population growth and decline
• Analyzing the relationship between two variables
• Solving problems in physics, engineering, and economics
• Understanding the behavior of linear systems

Q: How can we use the concept of comparing the graphs of two linear equations to solve problems?

A: We can use the concept of comparing the graphs of two linear equations to solve problems by:

• Identifying the slope and y-intercept of each equation
• Plotting the equations on the same coordinate plane
• Analyzing the relationship between the two graphs
• Using the information to solve the problem

Q: What are some common mistakes to avoid when comparing the graphs of two linear equations?

A: Some common mistakes to avoid when comparing the graphs of two linear equations include:

• Failing to identify the slope and y-intercept of each equation
• Plotting the equations on the same coordinate plane incorrectly
• Failing to analyze the relationship between the two graphs
• Not using the information to solve the problem

Q: How can we ensure that we are comparing the graphs of two linear equations correctly?

A: We can ensure that we are comparing the graphs of two linear equations correctly by:

• Double-checking the slope and y-intercept of each equation
• Plotting the equations on the same coordinate plane carefully
• Analyzing the relationship between the two graphs thoroughly
• Using the information to solve the problem accurately

Conclusion

In conclusion, comparing the graphs of two linear equations is an important concept in mathematics that has many real-world applications. By understanding how to compare the graphs of two linear equations, we can solve problems in physics, engineering, and economics, and analyze the relationship between two variables.