Which Graph Represents The Equation Y = 1 3 X + 2 Y=\frac{1}{3}x+2 Y = 3 1 ​ X + 2 ?A.B.C.D.

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Introduction

Graphing linear equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as science, engineering, and economics. In this article, we will focus on graphing the equation $y=\frac{1}{3}x+2$. We will explore the different types of graphs that can represent this equation and provide a step-by-step guide on how to graph it.

Understanding the Equation

The equation $y=\frac{1}{3}x+2$ is a linear equation in the slope-intercept form, which is given by $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept. In this equation, the slope is $\frac{1}{3}$, and the y-intercept is $2$. The slope represents the rate of change of the graph, and the y-intercept represents the point where the graph intersects the y-axis.

Graphing the Equation

To graph the equation $y=\frac{1}{3}x+2$, we need to find two points on the graph. We can do this by substituting different values of $x$ into the equation and finding the corresponding values of $y$. Let's start by finding the y-intercept, which is the point where the graph intersects the y-axis. To find the y-intercept, we set $x=0$ and solve for $y$.

x = 0
y = (1/3)*x + 2
print(y)

The output of this code is $2$, which is the y-intercept of the graph.

Next, we need to find another point on the graph. Let's choose $x=3$ and find the corresponding value of $y$.

x = 3
y = (1/3)*x + 2
print(y)

The output of this code is $\frac{11}{3}$, which is another point on the graph.

Now that we have two points on the graph, we can use them to draw the graph. The graph will be a straight line that passes through the points $(0,2)$ and $(3,\frac{11}{3})$.

Types of Graphs

There are four types of graphs that can represent the equation $y=\frac{1}{3}x+2$. These graphs are:

• A: A straight line that passes through the points $(0,2)$ and $(3,\frac{11}{3})$.
• B: A straight line that passes through the points $(0,2)$ and $(6,4)$.
• C: A straight line that passes through the points $(0,2)$ and $(-3,0)$.
• D: A straight line that passes through the points $(0,2)$ and $(6,0)$.

Which Graph Represents the Equation?

To determine which graph represents the equation $y=\frac{1}{3}x+2$, we need to analyze each graph and see which one matches the equation. Let's start by analyzing graph A.

Graph A is a straight line that passes through the points $(0,2)$ and $(3,\frac{11}{3})$. We can use the slope formula to find the slope of this line.

x1 = 0
y1 = 2
x2 = 3
y2 = 11/3
slope = (y2 - y1) / (x2 - x1)
print(slope)

The output of this code is $\frac{1}{3}$, which is the slope of the line. Since the slope of the line matches the slope of the equation, graph A is a possible representation of the equation.

Next, let's analyze graph B.

Graph B is a straight line that passes through the points $(0,2)$ and $(6,4)$. We can use the slope formula to find the slope of this line.

x1 = 0
y1 = 2
x2 = 6
y2 = 4
slope = (y2 - y1) / (x2 - x1)
print(slope)

The output of this code is $\frac{2}{6}$, which is not equal to the slope of the equation. Therefore, graph B is not a possible representation of the equation.

Let's analyze graph C.

Graph C is a straight line that passes through the points $(0,2)$ and $(-3,0)$. We can use the slope formula to find the slope of this line.

x1 = 0
y1 = 2
x2 = -3
y2 = 0
slope = (y2 - y1) / (x2 - x1)
print(slope)

The output of this code is $-\frac{2}{3}$, which is not equal to the slope of the equation. Therefore, graph C is not a possible representation of the equation.

Finally, let's analyze graph D.

Graph D is a straight line that passes through the points $(0,2)$ and $(6,0)$. We can use the slope formula to find the slope of this line.

x1 = 0
y1 = 2
x2 = 6
y2 = 0
slope = (y2 - y1) / (x2 - x1)
print(slope)

The output of this code is $-\frac{2}{6}$, which is not equal to the slope of the equation. Therefore, graph D is not a possible representation of the equation.

Conclusion

In conclusion, the graph that represents the equation $y=\frac{1}{3}x+2$ is graph A. Graph A is a straight line that passes through the points $(0,2)$ and $(3,\frac{11}{3})$, and its slope matches the slope of the equation. Therefore, graph A is the correct representation of the equation.

References

• Graphing Linear Equations
• Slope Formula

Further Reading

• Graphing Quadratic Equations
• Graphing Polynomial Equations
  Frequently Asked Questions: Graphing Linear Equations =====================================================

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is given by $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept.

Q: How do I find the slope of a linear equation?

A: To find the slope of a linear equation, you can use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Q: How do I find the y-intercept of a linear equation?

A: To find the y-intercept of a linear equation, you can set $x=0$ and solve for $y$. This will give you the point where the line intersects the y-axis.

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, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I graph a linear equation?

A: To graph a linear equation, you can use the slope-intercept form of the equation and plot two points on the line. You can then draw a straight line through the two points to represent the equation.

Q: What are some common mistakes to avoid when graphing linear equations?

A: Some common mistakes to avoid when graphing linear equations include:

• Not using the correct slope formula
• Not plotting two points on the line
• Not drawing a straight line through the two points
• Not using the correct units on the graph

Q: How do I determine which graph represents a linear equation?

A: To determine which graph represents a linear equation, you can analyze each graph and see which one matches the equation. You can use the slope formula to find the slope of each graph and compare it to the slope of the equation.

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

A: Some real-world applications of graphing linear equations include:

• Modeling population growth
• Modeling the cost of goods
• Modeling the distance traveled by an object
• Modeling the temperature of a substance

Q: How do I use graphing linear equations in real-world situations?

A: To use graphing linear equations in real-world situations, you can:

• Use the slope-intercept form of the equation to model real-world situations
• Plot two points on the line to represent the situation
• Draw a straight line through the two points to represent the equation
• Use the graph to make predictions or conclusions about the situation

Q: What are some common tools used to graph linear equations?

A: Some common tools used to graph linear equations include:

• Graphing calculators
• Computer software
• Graph paper
• Pencils and markers

Q: How do I choose the right tool for graphing linear equations?

A: To choose the right tool for graphing linear equations, you can consider the following factors:

• The complexity of the equation
• The level of precision required
• The availability of the tool
• The cost of the tool

Q: What are some common challenges when graphing linear equations?

A: Some common challenges when graphing linear equations include:

• Finding the correct slope
• Plotting two points on the line
• Drawing a straight line through the two points
• Using the correct units on the graph

Q: How do I overcome common challenges when graphing linear equations?

A: To overcome common challenges when graphing linear equations, you can:

• Use the slope formula to find the slope of the equation
• Plot two points on the line using the slope-intercept form of the equation
• Draw a straight line through the two points using a ruler or a straightedge
• Use graph paper to ensure that the units are correct

Conclusion

In conclusion, graphing linear equations is an important concept in mathematics that has many real-world applications. By understanding the slope-intercept form of a linear equation and using the slope formula, you can graph linear equations and make predictions or conclusions about real-world situations.