Which Graph Matches The Equation $y+6=\frac{3}{4}(x+4$\]?

Introduction

Graphing equations is a fundamental concept in mathematics, and it's essential to understand how to identify the correct graph that represents a given equation. In this article, we will explore the process of graphing the equation $y+6=\frac{3}{4}(x+4)$ and determine which graph matches this equation.

Understanding the Equation

The given equation is $y+6=\frac{3}{4}(x+4)$. To begin, we need to isolate the variable $y$ by subtracting $6$ from both sides of the equation. This gives us $y=\frac{3}{4}(x+4)-6$. We can simplify this expression by distributing the $\frac{3}{4}$ to the terms inside the parentheses, which results in $y=\frac{3}{4}x+3-6$. Combining the constants, we get $y=\frac{3}{4}x-3$.

Graphing the Equation

To graph the equation $y=\frac{3}{4}x-3$, we need to identify the key features of the graph. The equation is in the form of a linear equation, which means it will have a straight line as its graph. The slope of the line is $\frac{3}{4}$, which indicates that the line will have a positive slope. The y-intercept is $-3$, which means the line will intersect the y-axis at the point $(0,-3)$.

Identifying the Graph

Now that we have identified the key features of the graph, we can determine which graph matches the equation $y+6=\frac{3}{4}(x+4)$. To do this, we need to consider the transformations that have been applied to the graph. The equation $y+6=\frac{3}{4}(x+4)$ can be rewritten as $y=\frac{3}{4}(x+4)-6$. This means that the graph has been shifted up by $6$ units and to the left by $4$ units.

Graph A

Graph A is a straight line with a positive slope. However, it does not appear to have been shifted up or to the left. Therefore, Graph A does not match the equation $y+6=\frac{3}{4}(x+4)$.

Graph B

Graph B is a straight line with a positive slope. It appears to have been shifted up by $6$ units, but it has not been shifted to the left. Therefore, Graph B does not match the equation $y+6=\frac{3}{4}(x+4)$.

Graph C

Graph C is a straight line with a positive slope. It appears to have been shifted up by $6$ units and to the left by $4$ units. Therefore, Graph C matches the equation $y+6=\frac{3}{4}(x+4)$.

Conclusion

In conclusion, the graph that matches the equation $y+6=\frac{3}{4}(x+4)$ is Graph C. This graph has been shifted up by $6$ units and to the left by $4$ units, which is consistent with the transformations applied to the equation. Understanding how to identify the correct graph that represents a given equation is an essential skill in mathematics, and it requires a combination of algebraic and graphical skills.

Frequently Asked Questions

• Q: What is the slope of the graph that matches the equation $y+6=\frac{3}{4}(x+4)$? A: The slope of the graph is $\frac{3}{4}$.
• Q: What is the y-intercept of the graph that matches the equation $y+6=\frac{3}{4}(x+4)$? A: The y-intercept of the graph is $-3$.
• Q: What transformations have been applied to the graph that matches the equation $y+6=\frac{3}{4}(x+4)$? A: The graph has been shifted up by $6$ units and to the left by $4$ units.

Final Thoughts

Graphing equations is a fundamental concept in mathematics, and it's essential to understand how to identify the correct graph that represents a given equation. By following the steps outlined in this article, you can determine which graph matches the equation $y+6=\frac{3}{4}(x+4)$. Remember to identify the key features of the graph, including the slope and y-intercept, and consider the transformations that have been applied to the graph. With practice and experience, you will become proficient in graphing equations and identifying the correct graph that represents a given equation.

Introduction

Graphing equations is a fundamental concept in mathematics, and it's essential to understand how to identify the correct graph that represents a given equation. In our previous article, we explored the process of graphing the equation $y+6=\frac{3}{4}(x+4)$ and determined which graph matches this equation. In this article, we will answer some frequently asked questions about graphing equations and identifying the correct graph.

Q&A

Q: What is the first step in graphing an equation?

A: The first step in graphing an equation is to identify the key features of the graph, including the slope and y-intercept.

Q: How do I determine the slope of a graph?

A: To determine the slope of a graph, you need to look at the equation and identify the coefficient of the x-term. The slope is the ratio of the coefficient of the x-term to the coefficient of the y-term.

Q: What is the y-intercept of a graph?

A: The y-intercept of a graph is the point where the graph intersects the y-axis. It is the value of y when x is equal to 0.

Q: How do I identify the correct graph that represents a given equation?

A: To identify the correct graph that represents a given equation, you need to consider the transformations that have been applied to the graph. Look for shifts, stretches, and compressions that have been applied to the graph.

Q: What are some common transformations that are applied to graphs?

A: Some common transformations that are applied to graphs include:

• Shifts: Shifting the graph up or down by a certain number of units
• Stretches: Stretching the graph horizontally or vertically by a certain factor
• Compressions: Compressing the graph horizontally or vertically by a certain factor
• Reflections: Reflecting the graph across the x-axis or y-axis

Q: How do I graph an equation with multiple terms?

A: To graph an equation with multiple terms, you need to identify the key features of the graph, including the slope and y-intercept. You can then use these features to graph the equation.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. A quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you need to identify the key features of the graph, including the vertex and the x-intercepts. You can then use these features to graph the equation.

Conclusion

Graphing equations is a fundamental concept in mathematics, and it's essential to understand how to identify the correct graph that represents a given equation. By following the steps outlined in this article, you can answer some frequently asked questions about graphing equations and identifying the correct graph. Remember to identify the key features of the graph, including the slope and y-intercept, and consider the transformations that have been applied to the graph. With practice and experience, you will become proficient in graphing equations and identifying the correct graph that represents a given equation.

Frequently Asked Questions

• Q: What is the first step in graphing an equation? A: The first step in graphing an equation is to identify the key features of the graph, including the slope and y-intercept.
• Q: How do I determine the slope of a graph? A: To determine the slope of a graph, you need to look at the equation and identify the coefficient of the x-term. The slope is the ratio of the coefficient of the x-term to the coefficient of the y-term.
• Q: What is the y-intercept of a graph? A: The y-intercept of a graph is the point where the graph intersects the y-axis. It is the value of y when x is equal to 0.

Final Thoughts

Graphing equations is a fundamental concept in mathematics, and it's essential to understand how to identify the correct graph that represents a given equation. By following the steps outlined in this article, you can answer some frequently asked questions about graphing equations and identifying the correct graph. Remember to identify the key features of the graph, including the slope and y-intercept, and consider the transformations that have been applied to the graph. With practice and experience, you will become proficient in graphing equations and identifying the correct graph that represents a given equation.