Sketch A Graph And Identify The Vertex For The Equation: Y = ∣ X + 2 ∣ − 4 Y = |x + 2| - 4 Y = ∣ X + 2∣ − 4

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Introduction

In mathematics, graphing equations is an essential skill that helps us visualize the relationship between variables. When dealing with absolute value equations, it's crucial to understand how the absolute value function affects the graph. In this article, we will explore the graph of the equation $y = |x + 2| - 4$ and identify its vertex.

Understanding Absolute Value Equations

Absolute value equations involve the absolute value of a quantity, which is always non-negative. The absolute value of a number $x$ is denoted by $|x|$ and is defined as:

$$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$

When we have an absolute value equation of the form $y = |x - a| + b$, the graph of the equation is a V-shaped graph with its vertex at the point $(a, b)$.

Graphing the Equation $y = |x + 2| - 4$

To graph the equation $y = |x + 2| - 4$, we need to consider two cases:

Case 1: $x \geq -2$

When $x \geq -2$, the expression $x + 2$ is non-negative, and the absolute value function can be rewritten as:

$$y = x + 2 - 4 = x - 2$$

This is a linear equation with a slope of 1 and a y-intercept of -2. The graph of this equation is a straight line with a slope of 1 and a y-intercept of -2.

Case 2: $x < -2$

When $x < -2$, the expression $x + 2$ is negative, and the absolute value function can be rewritten as:

$$y = -(x + 2) - 4 = -x - 6$$

This is also a linear equation with a slope of -1 and a y-intercept of -6. The graph of this equation is a straight line with a slope of -1 and a y-intercept of -6.

Combining the Two Cases

To obtain the graph of the equation $y = |x + 2| - 4$, we need to combine the two cases. When $x \geq -2$, the graph is a straight line with a slope of 1 and a y-intercept of -2. When $x < -2$, the graph is a straight line with a slope of -1 and a y-intercept of -6.

The graph of the equation $y = |x + 2| - 4$ is a V-shaped graph with its vertex at the point $(-2, -4)$.

Identifying the Vertex

The vertex of the graph of the equation $y = |x + 2| - 4$ is the point where the graph changes from decreasing to increasing. In this case, the vertex is at the point $(-2, -4)$.

To identify the vertex, we need to find the x-coordinate of the vertex, which is the value of $x$ that makes the expression $x + 2$ equal to 0. In this case, the x-coordinate of the vertex is $-2$.

The y-coordinate of the vertex is the value of $y$ that corresponds to the x-coordinate of the vertex. In this case, the y-coordinate of the vertex is $-4$.

Therefore, the vertex of the graph of the equation $y = |x + 2| - 4$ is the point $(-2, -4)$.

Conclusion

In this article, we have explored the graph of the equation $y = |x + 2| - 4$ and identified its vertex. We have seen that the graph of the equation is a V-shaped graph with its vertex at the point $(-2, -4)$. We have also seen that the vertex of the graph is the point where the graph changes from decreasing to increasing.

Key Takeaways

• The graph of the equation $y = |x + 2| - 4$ is a V-shaped graph with its vertex at the point $(-2, -4)$.
• The vertex of the graph is the point where the graph changes from decreasing to increasing.
• The x-coordinate of the vertex is the value of $x$ that makes the expression $x + 2$ equal to 0.
• The y-coordinate of the vertex is the value of $y$ that corresponds to the x-coordinate of the vertex.

Practice Problems

1. Graph the equation $y = |x - 3| + 2$ and identify its vertex.
2. Graph the equation $y = |x + 1| - 1$ and identify its vertex.
3. Graph the equation $y = |x - 2| + 1$ and identify its vertex.

Solutions

1. The graph of the equation $y = |x - 3| + 2$ is a V-shaped graph with its vertex at the point $(3, 2)$.
2. The graph of the equation $y = |x + 1| - 1$ is a V-shaped graph with its vertex at the point $(-1, -1)$.
3. The graph of the equation $y = |x - 2| + 1$ is a V-shaped graph with its vertex at the point $(2, 1)$.
   Q&A: Graphing Absolute Value Equations and Identifying Vertices ================================================================

Frequently Asked Questions

Q: What is the graph of the equation $y = |x + 2| - 4$?

A: The graph of the equation $y = |x + 2| - 4$ is a V-shaped graph with its vertex at the point $(-2, -4)$.

Q: How do I identify the vertex of the graph of an absolute value equation?

A: To identify the vertex of the graph of an absolute value equation, you need to find the x-coordinate of the vertex, which is the value of $x$ that makes the expression inside the absolute value equal to 0. Then, you need to find the y-coordinate of the vertex, which is the value of $y$ that corresponds to the x-coordinate of the vertex.

Q: What is the x-coordinate of the vertex of the graph of the equation $y = |x + 2| - 4$?

A: The x-coordinate of the vertex of the graph of the equation $y = |x + 2| - 4$ is $-2$.

Q: What is the y-coordinate of the vertex of the graph of the equation $y = |x + 2| - 4$?

A: The y-coordinate of the vertex of the graph of the equation $y = |x + 2| - 4$ is $-4$.

Q: How do I graph the equation $y = |x - 3| + 2$?

A: To graph the equation $y = |x - 3| + 2$, you need to consider two cases: when $x \geq 3$ and when $x < 3$. When $x \geq 3$, the graph is a straight line with a slope of 1 and a y-intercept of 5. When $x < 3$, the graph is a straight line with a slope of -1 and a y-intercept of -1.

Q: What is the vertex of the graph of the equation $y = |x - 3| + 2$?

A: The vertex of the graph of the equation $y = |x - 3| + 2$ is the point $(3, 2)$.

Q: How do I graph the equation $y = |x + 1| - 1$?

A: To graph the equation $y = |x + 1| - 1$, you need to consider two cases: when $x \geq -1$ and when $x < -1$. When $x \geq -1$, the graph is a straight line with a slope of 1 and a y-intercept of -2. When $x < -1$, the graph is a straight line with a slope of -1 and a y-intercept of -2.

Q: What is the vertex of the graph of the equation $y = |x + 1| - 1$?

A: The vertex of the graph of the equation $y = |x + 1| - 1$ is the point $(-1, -1)$.

Q: How do I graph the equation $y = |x - 2| + 1$?

A: To graph the equation $y = |x - 2| + 1$, you need to consider two cases: when $x \geq 2$ and when $x < 2$. When $x \geq 2$, the graph is a straight line with a slope of 1 and a y-intercept of 3. When $x < 2$, the graph is a straight line with a slope of -1 and a y-intercept of -1.

Q: What is the vertex of the graph of the equation $y = |x - 2| + 1$?

A: The vertex of the graph of the equation $y = |x - 2| + 1$ is the point $(2, 1)$.

Additional Resources

• For more information on graphing absolute value equations, see the article "Graphing Absolute Value Equations".
• For more information on identifying vertices, see the article "Identifying Vertices of Graphs".
• For practice problems and solutions, see the article "Practice Problems and Solutions".

Conclusion

In this article, we have answered frequently asked questions about graphing absolute value equations and identifying vertices. We have seen that the graph of an absolute value equation is a V-shaped graph with its vertex at the point where the graph changes from decreasing to increasing. We have also seen that the x-coordinate of the vertex is the value of $x$ that makes the expression inside the absolute value equal to 0, and the y-coordinate of the vertex is the value of $y$ that corresponds to the x-coordinate of the vertex.