Graph The Equation:$ Y = -2|x+5| - 3 $

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Understanding the Equation

The given equation is a linear function in the form of $y = a|x+b| + c$, where $a$, $b$, and $c$ are constants. In this case, $a = -2$, $b = 5$, and $c = -3$. This type of equation is known as a linear absolute value function.

Graphing Absolute Value Functions

To graph an absolute value function, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: When $x+5 \geq 0$

When $x+5 \geq 0$, the expression inside the absolute value is positive, and the equation becomes $y = -2(x+5) - 3$. This is a linear function with a slope of $-2$ and a y-intercept of $-13$.

Case 2: When $x+5 < 0$

When $x+5 < 0$, the expression inside the absolute value is negative, and the equation becomes $y = -2(-(x+5)) - 3$. This is also a linear function with a slope of $2$ and a y-intercept of $-13$.

Graphing the Equation

To graph the equation $y = -2|x+5| - 3$, we need to consider both cases and graph the corresponding linear functions.

Graphing the Linear Function for Case 1

For case 1, when $x+5 \geq 0$, the linear function is $y = -2(x+5) - 3$. This is a line with a slope of $-2$ and a y-intercept of $-13$. The graph of this line is a downward-sloping line that passes through the point $(-5, -13)$.

Graphing the Linear Function for Case 2

For case 2, when $x+5 < 0$, the linear function is $y = -2(-(x+5)) - 3$. This is also a line with a slope of $2$ and a y-intercept of $-13$. The graph of this line is an upward-sloping line that passes through the point $(-5, -13)$.

Combining the Graphs

To graph the equation $y = -2|x+5| - 3$, we need to combine the graphs of the two linear functions. The resulting graph is a V-shaped graph with a vertex at $(-5, -13)$.

Key Features of the Graph

The graph of the equation $y = -2|x+5| - 3$ has the following key features:

• Vertex: The vertex of the graph is at $(-5, -13)$.
• Axis of Symmetry: The axis of symmetry of the graph is the vertical line $x = -5$.
• Slope: The slope of the graph is $-2$ for $x+5 \geq 0$ and $2$ for $x+5 < 0$.
• Y-Intercept: The y-intercept of the graph is $-13$.

Graphing the Equation Using Technology

To graph the equation $y = -2|x+5| - 3$ using technology, we can use a graphing calculator or a computer algebra system (CAS). These tools can help us visualize the graph and identify its key features.

Conclusion

In conclusion, the graph of the equation $y = -2|x+5| - 3$ is a V-shaped graph with a vertex at $(-5, -13)$. The graph has a slope of $-2$ for $x+5 \geq 0$ and $2$ for $x+5 < 0$, and a y-intercept of $-13$. We can graph the equation using technology or by hand, and identify its key features.

Applications of the Equation

The equation $y = -2|x+5| - 3$ has several applications in mathematics and real-world problems. Some examples include:

• Modeling Real-World Problems: The equation can be used to model real-world problems that involve absolute value functions, such as the distance between two points or the cost of a product.
• Graphing Functions: The equation can be used to graph functions that involve absolute value, such as the graph of a linear function with a slope of $-2$.
• Solving Equations: The equation can be used to solve equations that involve absolute value, such as the equation $|x+5| = 3$.

Tips for Graphing the Equation

Here are some tips for graphing the equation $y = -2|x+5| - 3$:

• Use a Graphing Calculator: A graphing calculator can help you visualize the graph and identify its key features.
• Use a Computer Algebra System (CAS): A CAS can help you graph the equation and identify its key features.
• Graph the Equation by Hand: You can graph the equation by hand by using the two cases and graphing the corresponding linear functions.
• Identify the Key Features: Once you have graphed the equation, identify its key features, such as the vertex, axis of symmetry, slope, and y-intercept.

Common Mistakes to Avoid

Here are some common mistakes to avoid when graphing the equation $y = -2|x+5| - 3$:

• Not Considering Both Cases: Make sure to consider both cases when graphing the equation.
• Not Graphing the Correct Linear Function: Make sure to graph the correct linear function for each case.
• Not Identifying the Key Features: Make sure to identify the key features of the graph, such as the vertex, axis of symmetry, slope, and y-intercept.

Conclusion

In conclusion, the graph of the equation $y = -2|x+5| - 3$ is a V-shaped graph with a vertex at $(-5, -13)$. The graph has a slope of $-2$ for $x+5 \geq 0$ and $2$ for $x+5 < 0$, and a y-intercept of $-13$. We can graph the equation using technology or by hand, and identify its key features.

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Understanding the Equation

The given equation is a linear function in the form of $y = a|x+b| + c$, where $a$, $b$, and $c$ are constants. In this case, $a = -2$, $b = 5$, and $c = -3$. This type of equation is known as a linear absolute value function.

Q&A

Q: What is the vertex of the graph?

A: The vertex of the graph is at $(-5, -13)$.

Q: What is the axis of symmetry of the graph?

A: The axis of symmetry of the graph is the vertical line $x = -5$.

Q: What is the slope of the graph?

A: The slope of the graph is $-2$ for $x+5 \geq 0$ and $2$ for $x+5 < 0$.

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

A: The y-intercept of the graph is $-13$.

Q: How do I graph the equation using technology?

A: You can use a graphing calculator or a computer algebra system (CAS) to graph the equation.

Q: How do I graph the equation by hand?

A: You can graph the equation by hand by using the two cases and graphing the corresponding linear functions.

Q: What are some common mistakes to avoid when graphing the equation?

A: Some common mistakes to avoid when graphing the equation include not considering both cases, not graphing the correct linear function, and not identifying the key features of the graph.

Q: What are some applications of the equation?

A: The equation has several applications in mathematics and real-world problems, including modeling real-world problems, graphing functions, and solving equations.

Q: How do I identify the key features of the graph?

A: To identify the key features of the graph, you need to consider the vertex, axis of symmetry, slope, and y-intercept.

Q: What is the significance of the axis of symmetry?

A: The axis of symmetry is the vertical line that passes through the vertex of the graph and divides the graph into two equal parts.

Q: How do I use the equation to model real-world problems?

A: You can use the equation to model real-world problems that involve absolute value functions, such as the distance between two points or the cost of a product.

Q: How do I use the equation to graph functions?

A: You can use the equation to graph functions that involve absolute value, such as the graph of a linear function with a slope of $-2$.

Q: How do I use the equation to solve equations?

A: You can use the equation to solve equations that involve absolute value, such as the equation $|x+5| = 3$.

Conclusion

In conclusion, the graph of the equation $y = -2|x+5| - 3$ is a V-shaped graph with a vertex at $(-5, -13)$. The graph has a slope of $-2$ for $x+5 \geq 0$ and $2$ for $x+5 < 0$, and a y-intercept of $-13$. We can graph the equation using technology or by hand, and identify its key features.