The function $h(x)=\sqrt{2 x-8}+1$ is a transformation of the parent function $f(x)=\sqrt{x}$. Select all the statements that describe a key feature of function $h$.

The function h ( x ) = 2 x − 8 ​ + 1 is a transformation of f ( x ) = x ​ .
h ( x ) has a horizontal compression by a factor of 2 1 ​ , a horizontal translation 4 units to the right, and a vertical translation 1 unit upwards.
The domain of h ( x ) is x ≥ 4 .
The range of h ( x ) is h ( x ) ≥ 1 .
The key features of function h are horizontal compression, horizontal and vertical translations, domain, and range. h ( x ) = 2 x − 8 ​ + 1 ​

Explanation

Understanding the Problem The function h ( x ) = 2 x − 8 ​ + 1 is a transformation of the parent function f ( x ) = x ​ . We need to identify the key features of the transformed function h ( x ) .

Rewriting the Function First, let's rewrite the function h ( x ) to clearly identify the transformations applied to the parent function f ( x ) = x ​ . We can rewrite h ( x ) as follows:


h ( x ) = 2 ( x − 4 ) ​ + 1

Identifying Transformations Now, let's identify the transformations:

Horizontal Compression: The factor of 2 inside the square root causes a horizontal compression by a factor of 2 1 ​ .

Horizontal Translation: The term ( x − 4 ) indicates a horizontal translation of 4 units to the right.

Vertical Translation: The '+1' outside the square root indicates a vertical translation of 1 unit upwards.

Determining the Domain Let's determine the domain of h ( x ) . Since the expression inside the square root must be non-negative, we have:


2 x − 8 ≥ 0
2 x ≥ 8
x ≥ 4
So, the domain of h ( x ) is x ≥ 4 .

Determining the Range Now, let's determine the range of h ( x ) . Since the square root function always returns a non-negative value, 2 x − 8 ​ ≥ 0 . Therefore:

h ( x ) = 2 x − 8 ​ + 1 ≥ 1
So, the range of h ( x ) is h ( x ) ≥ 1 .

Concluding Key Features Based on the transformations and the domain/range, we can now select the statements that accurately describe the key features of function h .

Key features of h ( x ) :

Horizontal compression by a factor of 2 1 ​ .
Horizontal translation 4 units to the right.
Vertical translation 1 unit upwards.
Domain: x ≥ 4
Range: h ( x ) ≥ 1

Examples
Understanding transformations of functions is crucial in various fields. For instance, in physics, understanding how graphs of motion change with different parameters (like time or initial velocity) helps predict the behavior of objects. Similarly, in economics, transformations of supply and demand curves can model the impact of taxes or subsidies on market equilibrium. This concept is also used in image processing, where transformations like scaling, shifting, and rotating are applied to images to achieve desired effects or correct distortions. For example, the function h ( x ) = 2 f ( x − 3 ) + 1 represents a transformation of f ( x ) that includes a horizontal shift, a vertical stretch, and a vertical shift, all of which can be visualized and understood through graphical transformations.

Answered by GinnyAnswer | 2025-07-03

The function h ( x ) = 2 x − 8 ​ + 1 undergoes a horizontal compression by a factor of 2 1 ​ , a horizontal translation of 4 units to the right, and a vertical translation of 1 unit upwards. Its domain is x ≥ 4 and its range is h ( x ) ≥ 1 .
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Answered by Anonymous | 2025-07-04