Given $g(x)=|x-3|$, Complete The Statements:The Vertex Is: $(\square , \square$\].The Axis Of Symmetry Is: $x= \square$.

by ADMIN 123 views

Introduction

In mathematics, absolute value functions are a type of function that involves the absolute value of a variable. These functions are commonly used in various mathematical applications, including graphing and solving equations. In this article, we will focus on the graph of a linear absolute value function, specifically the function g(x)=∣x−3∣g(x) = |x - 3|. We will complete the statements about the vertex and axis of symmetry of this function.

The Graph of a Linear Absolute Value Function

A linear absolute value function is a function of the form f(x)=∣ax+b∣f(x) = |ax + b|, where aa and bb are constants. The graph of this function is a V-shaped graph with its vertex at the point (h,k)(h, k), where h=−bah = -\frac{b}{a} and k=∣ah+b∣k = |ah + b|.

The Vertex of the Graph

The vertex of the graph of a linear absolute value function is the point at which the graph changes direction. To find the vertex of the graph of g(x)=∣x−3∣g(x) = |x - 3|, we need to find the value of xx at which the graph changes direction.

Finding the Vertex

To find the vertex of the graph of g(x)=∣x−3∣g(x) = |x - 3|, we need to find the value of xx at which the graph changes direction. This occurs when the expression inside the absolute value bars is equal to zero.

x−3=0x - 3 = 0

Solving for xx, we get:

x=3x = 3

Therefore, the vertex of the graph of g(x)=∣x−3∣g(x) = |x - 3| is at the point (3,0)(3, 0).

The Vertex is: (□,□)(\square , \square)

The vertex of the graph of g(x)=∣x−3∣g(x) = |x - 3| is at the point (3,0)(3, 0). Therefore, the completed statement is:

The vertex is: (3,0)(3, 0).

The Axis of Symmetry

The axis of symmetry of a graph is a vertical line that passes through the vertex of the graph. To find the axis of symmetry of the graph of g(x)=∣x−3∣g(x) = |x - 3|, we need to find the equation of the vertical line that passes through the vertex of the graph.

Finding the Axis of Symmetry

To find the axis of symmetry of the graph of g(x)=∣x−3∣g(x) = |x - 3|, we need to find the equation of the vertical line that passes through the vertex of the graph. Since the vertex of the graph is at the point (3,0)(3, 0), the equation of the vertical line that passes through this point is:

x=3x = 3

Therefore, the axis of symmetry of the graph of g(x)=∣x−3∣g(x) = |x - 3| is the vertical line x=3x = 3.

The Axis of Symmetry is: x=□x= \square

The axis of symmetry of the graph of g(x)=∣x−3∣g(x) = |x - 3| is the vertical line x=3x = 3. Therefore, the completed statement is:

The axis of symmetry is: x=3x = 3.

Conclusion

In this article, we have completed the statements about the vertex and axis of symmetry of the graph of the linear absolute value function g(x)=∣x−3∣g(x) = |x - 3|. We have found that the vertex of the graph is at the point (3,0)(3, 0) and the axis of symmetry is the vertical line x=3x = 3. This information can be useful in graphing and solving equations involving linear absolute value functions.

References

Additional Resources

Introduction

In our previous article, we discussed the graph of a linear absolute value function, specifically the function g(x)=∣x−3∣g(x) = |x - 3|. We completed the statements about the vertex and axis of symmetry of this function. In this article, we will answer some frequently asked questions about linear absolute value functions.

Q: What is a linear absolute value function?

A linear absolute value function is a function of the form f(x)=∣ax+b∣f(x) = |ax + b|, where aa and bb are constants. The graph of this function is a V-shaped graph with its vertex at the point (h,k)(h, k), where h=−bah = -\frac{b}{a} and k=∣ah+b∣k = |ah + b|.

Q: How do I find the vertex of a linear absolute value function?

To find the vertex of a linear absolute value function, you need to find the value of xx at which the graph changes direction. This occurs when the expression inside the absolute value bars is equal to zero.

Q: What is the axis of symmetry of a linear absolute value function?

The axis of symmetry of a linear absolute value function is a vertical line that passes through the vertex of the graph. To find the axis of symmetry, you need to find the equation of the vertical line that passes through the vertex of the graph.

Q: How do I graph a linear absolute value function?

To graph a linear absolute value function, you need to follow these steps:

  1. Find the vertex of the graph.
  2. Find the axis of symmetry of the graph.
  3. Plot the vertex and the axis of symmetry on a coordinate plane.
  4. Plot two points on either side of the axis of symmetry, one above and one below the vertex.
  5. Connect the points with a V-shaped graph.

Q: What are some common applications of linear absolute value functions?

Linear absolute value functions have many applications in mathematics and real-world problems. Some common applications include:

  • Modeling the distance between two points on a coordinate plane.
  • Finding the maximum or minimum value of a function.
  • Solving equations involving absolute value.

Q: How do I solve equations involving absolute value?

To solve equations involving absolute value, you need to follow these steps:

  1. Isolate the absolute value expression on one side of the equation.
  2. Set the expression inside the absolute value bars equal to zero and solve for xx.
  3. Set the expression inside the absolute value bars equal to a positive value and solve for xx.
  4. Set the expression inside the absolute value bars equal to a negative value and solve for xx.
  5. Check your solutions to make sure they are valid.

Q: What are some common mistakes to avoid when working with linear absolute value functions?

Some common mistakes to avoid when working with linear absolute value functions include:

  • Not following the order of operations when simplifying expressions.
  • Not checking for extraneous solutions when solving equations.
  • Not using the correct notation when writing equations.

Conclusion

In this article, we have answered some frequently asked questions about linear absolute value functions. We have discussed the graph of a linear absolute value function, how to find the vertex and axis of symmetry, and how to graph and solve equations involving absolute value. We have also discussed some common applications and mistakes to avoid when working with linear absolute value functions.

References

Additional Resources