Graph The Following Absolute Value Expression:$ Y = |x - 4| - 3 }$Show Your Work Here 1. Identify The Vertex Of The Graph. The Expression Inside The Absolute Value, { X - 4 $ $, Equals Zero When { X = 4 $}$.

Understanding Absolute Value Expressions

Absolute value expressions are a fundamental concept in mathematics, and graphing them can be a bit tricky. In this article, we will explore how to graph the absolute value expression $y = |x - 4| - 3$ and identify the vertex of the graph.

Step 1: Identify the Vertex of the Graph

The expression inside the absolute value, $x - 4$, equals zero when $x = 4$. This means that the vertex of the graph is at the point $(4, -3)$.

Step 2: Graph the Absolute Value Expression

To graph the absolute value expression, we need to consider two cases:

• When $x < 4$, the expression inside the absolute value is negative, so we have $y = -(x - 4) - 3$.
• When $x \geq 4$, the expression inside the absolute value is non-negative, so we have $y = x - 4 - 3$.

Graphing the First Case: $x < 4$

When $x < 4$, the expression inside the absolute value is negative, so we have $y = -(x - 4) - 3$. To graph this expression, we can start by finding the x-intercept, which occurs when $y = 0$. Setting $y = 0$, we get:

$$0 = -(x - 4) - 3$$

Solving for $x$, we get:

$$x - 4 = 3$$

$$x = 7$$

So, the x-intercept is at the point $(7, 0)$.

Graphing the Second Case: $x \geq 4$

When $x \geq 4$, the expression inside the absolute value is non-negative, so we have $y = x - 4 - 3$. To graph this expression, we can start by finding the x-intercept, which occurs when $y = 0$. Setting $y = 0$, we get:

$$0 = x - 4 - 3$$

Solving for $x$, we get:

$$x = 7$$

So, the x-intercept is at the point $(7, 0)$.

Combining the Two Cases

Now that we have graphed the two cases, we can combine them to get the complete graph of the absolute value expression.

The Complete Graph

The complete graph of the absolute value expression $y = |x - 4| - 3$ is a V-shaped graph with its vertex at the point $(4, -3)$. The graph has a minimum value of $-3$ at $x = 4$ and opens upwards on both sides.

Key Features of the Graph

• Vertex: The vertex of the graph is at the point $(4, -3)$.
• Minimum Value: The minimum value of the graph is $-3$ at $x = 4$.
• Opening: The graph opens upwards on both sides.

Conclusion

Graphing absolute value expressions can be a bit tricky, but by breaking it down into two cases and identifying the vertex, we can create a complete graph. In this article, we graphed the absolute value expression $y = |x - 4| - 3$ and identified the vertex of the graph. We also discussed the key features of the graph, including the vertex, minimum value, and opening.

Frequently Asked Questions

• What is the vertex of the graph?
  • The vertex of the graph is at the point $(4, -3)$.
• What is the minimum value of the graph?
  • The minimum value of the graph is $-3$ at $x = 4$.
• How does the graph open?
  • The graph opens upwards on both sides.

References

• Graphing Absolute Value Functions
• Absolute Value Functions

Additional Resources

• Graphing Absolute Value Functions
• Absolute Value Functions
  Graphing Absolute Value Expressions: A Q&A Guide =====================================================

Frequently Asked Questions

Q: What is the vertex of the graph of the absolute value expression $y = |x - 4| - 3$?

A: The vertex of the graph is at the point $(4, -3)$.

Q: How do I graph the absolute value expression $y = |x - 4| - 3$?

A: To graph the absolute value expression, you need to consider two cases:

• When $x < 4$, the expression inside the absolute value is negative, so you have $y = -(x - 4) - 3$.
• When $x \geq 4$, the expression inside the absolute value is non-negative, so you have $y = x - 4 - 3$.

Q: What is the minimum value of the graph of the absolute value expression $y = |x - 4| - 3$?

A: The minimum value of the graph is $-3$ at $x = 4$.

Q: How does the graph of the absolute value expression $y = |x - 4| - 3$ open?

A: The graph opens upwards on both sides.

Q: What is the x-intercept of the graph of the absolute value expression $y = |x - 4| - 3$?

A: The x-intercept is at the point $(7, 0)$.

Q: How do I find the x-intercept of the graph of the absolute value expression $y = |x - 4| - 3$?

A: To find the x-intercept, you need to set $y = 0$ and solve for $x$. For the first case, you get $x = 7$, and for the second case, you also get $x = 7$.

Q: What is the y-intercept of the graph of the absolute value expression $y = |x - 4| - 3$?

A: The y-intercept is at the point $(0, -3)$.

Q: How do I find the y-intercept of the graph of the absolute value expression $y = |x - 4| - 3$?

A: To find the y-intercept, you need to set $x = 0$ and solve for $y$. You get $y = -3$.

Q: Can I graph the absolute value expression $y = |x - 4| - 3$ using a graphing calculator?

A: Yes, you can graph the absolute value expression using a graphing calculator. Simply enter the expression into the calculator and graph it.

Q: How do I graph the absolute value expression $y = |x - 4| - 3$ using a graphing software?

A: You can graph the absolute value expression using a graphing software such as Desmos or GeoGebra. Simply enter the expression into the software and graph it.

Q: Can I graph the absolute value expression $y = |x - 4| - 3$ using a pencil and paper?

A: Yes, you can graph the absolute value expression using a pencil and paper. Simply draw the graph by hand, using the two cases to determine the shape of the graph.

Conclusion

Graphing absolute value expressions can be a bit tricky, but by breaking it down into two cases and identifying the vertex, we can create a complete graph. In this article, we answered some frequently asked questions about graphing absolute value expressions, including the vertex, minimum value, and opening of the graph. We also discussed how to find the x-intercept and y-intercept of the graph.

Frequently Asked Questions (FAQs)

• What is the vertex of the graph of the absolute value expression $y = |x - 4| - 3$?
  • The vertex of the graph is at the point $(4, -3)$.
• How do I graph the absolute value expression $y = |x - 4| - 3$?
  • To graph the absolute value expression, you need to consider two cases: when $x < 4$ and when $x \geq 4$.
• What is the minimum value of the graph of the absolute value expression $y = |x - 4| - 3$?
  • The minimum value of the graph is $-3$ at $x = 4$.
• How does the graph of the absolute value expression $y = |x - 4| - 3$ open?
  • The graph opens upwards on both sides.

References

• Graphing Absolute Value Functions
• Absolute Value Functions

Additional Resources

• Graphing Absolute Value Functions
• Absolute Value Functions