Understanding the Parent Function

Oct 13, 2025 by pynkocean.com 95 views

Understanding the Parent Function

The given function is ${ h(x) = \frac{1}{2}|x-2| - 3 }$ . To graph this function, we need to understand its parent function and the transformations applied to it. The parent function of the absolute value function is ${ f(x) = |x| }$ , which is a V-shaped graph with its vertex at the origin.

Identifying the Transformations

The given function can be broken down into three main transformations:

Vertical Stretch: The coefficient ${ \frac{1}{2} }$ in front of the absolute value function indicates a vertical stretch by a factor of ${ \frac{1}{2} }$ .
Horizontal Shift: The term ${ x-2 }$ inside the absolute value function indicates a horizontal shift of 2 units to the right.
Vertical Shift: The term ${ -3 }$ at the end of the function indicates a vertical shift of 3 units down.

Graphing the Function

To graph the function, we will apply these transformations to the parent function ${ f(x) = |x| }$ .

Step 1: Apply the Vertical Stretch

The vertical stretch by a factor of ${ \frac{1}{2} }$ will make the V-shaped graph narrower.

**Parent Function:** f(x) = |x|

**After Vertical Stretch:** f(x) = (1/2)|x|

Step 2: Apply the Horizontal Shift

The horizontal shift of 2 units to the right will move the V-shaped graph to the right.

**After Horizontal Shift:** f(x) = (1/2)|x-2|

Step 3: Apply the Vertical Shift

The vertical shift of 3 units down will move the V-shaped graph down.

**After Vertical Shift:** h(x) = (1/2)|x-2| - 3

Graphing the Function

Now that we have applied all the transformations, we can graph the function ${ h(x) = \frac{1}{2}|x-2| - 3 }$ .

The graph of the function will be a V-shaped graph with its vertex at ${ (2, -3) }$ . The graph will be narrower than the parent function due to the vertical stretch, and it will be shifted 2 units to the right and 3 units down.

Key Features of the Graph

Vertex: The vertex of the graph is at ${ (2, -3) }$ .
Axis of Symmetry: The axis of symmetry is the vertical line ${ x = 2 }$ .
Intercepts: The graph has no x-intercepts, but it has a y-intercept at ${ (0, -3) }$ .

Conclusion

Graphing the function ${ h(x) = \frac{1}{2}|x-2| - 3 }$ requires understanding the parent function and the transformations applied to it. By applying the vertical stretch, horizontal shift, and vertical shift, we can graph the function and identify its key features.

Practice Problems

Graph the function ${ f(x) = \frac{1}{3}|x-4| + 2 }$ .
Identify the vertex, axis of symmetry, and intercepts of the graph of the function ${ g(x) = 2|x-1| - 1 }$ .

Solutions

The graph of the function ${ f(x) = \frac{1}{3}|x-4| + 2 }$ will be a V-shaped graph with its vertex at ${ (4, 2) }$ . The graph will be narrower than the parent function due to the vertical stretch, and it will be shifted 4 units to the right and 2 units up.
The vertex of the graph of the function ${ g(x) = 2|x-1| - 1 }$ is at ${ (1, -1) }$ . The axis of symmetry is the vertical line ${ x = 1 }$ . The graph has no x-intercepts, but it has a y-intercept at ${ (0, -1) }$ .
Graphing the Function: A Comprehensive Guide =====================================================

Q&A: Graphing the Function

Q: What is the parent function of the absolute value function?

A: The parent function of the absolute value function is ${ f(x) = |x| }$ , which is a V-shaped graph with its vertex at the origin.

Q: What are the three main transformations applied to the parent function?

A: The three main transformations applied to the parent function are:

Vertical Stretch: The coefficient ${ \frac{1}{2} }$ in front of the absolute value function indicates a vertical stretch by a factor of ${ \frac{1}{2} }$ .
Horizontal Shift: The term ${ x-2 }$ inside the absolute value function indicates a horizontal shift of 2 units to the right.
Vertical Shift: The term ${ -3 }$ at the end of the function indicates a vertical shift of 3 units down.

Q: How do I graph the function ${ h(x) = \frac{1}{2}|x-2| - 3 }$ ?

A: To graph the function, you need to apply the three main transformations to the parent function ${ f(x) = |x| }$ . First, apply the vertical stretch by a factor of ${ \frac{1}{2} }$ , then apply the horizontal shift of 2 units to the right, and finally apply the vertical shift of 3 units down.

Q: What are the key features of the graph of the function ${ h(x) = \frac{1}{2}|x-2| - 3 }$ ?

A: The key features of the graph of the function are:

Vertex: The vertex of the graph is at ${ (2, -3) }$ .
Axis of Symmetry: The axis of symmetry is the vertical line ${ x = 2 }$ .
Intercepts: The graph has no x-intercepts, but it has a y-intercept at ${ (0, -3) }$ .

Q: How do I identify the vertex, axis of symmetry, and intercepts of the graph of the function?

A: To identify the vertex, axis of symmetry, and intercepts of the graph of the function, you need to analyze the function and apply the transformations to the parent function. The vertex will be at the point where the function changes from increasing to decreasing or vice versa. The axis of symmetry will be the vertical line that passes through the vertex. The intercepts will be the points where the graph intersects the x-axis and y-axis.

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

A: Some common mistakes to avoid when graphing the function are:

Incorrectly applying the transformations: Make sure to apply the transformations in the correct order and with the correct coefficients.
Not identifying the vertex, axis of symmetry, and intercepts: Make sure to analyze the function and identify the key features of the graph.
Not graphing the function accurately: Make sure to graph the function accurately and label the key features.

Q: How can I practice graphing the function?

A: You can practice graphing the function by:

Graphing multiple functions: Graph multiple functions with different transformations to practice identifying the key features.
Using graphing software: Use graphing software to graph the function and analyze the key features.
Working with a partner: Work with a partner to graph the function and discuss the key features.