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Chapter 0: Problem 41

Use transformations to graph each function. $$ y=\sqrt{x-1}+2 $$

Short Answer

Expert verified
The graph of \( y = \sqrt{x-1} + 2 \) is the graph of \( y = \sqrt{x} \) shifted 1 unit to the right and 2 units up.

Step by step solution

01

Identify the Parent Function

The parent function for the given equation is the square root function: \( y = \sqrt{x} \).
02

Horizontal Shift

The term \( x-1 \) inside the square root indicates a horizontal shift. Since it is \( x-1 \), the graph shifts 1 unit to the right.
03

Vertical Shift

The constant term +2 outside the square root function indicates a vertical shift. Therefore, the graph moves 2 units upward.
04

Graph the Transformed Function

Start with the basic shape of \( y = \sqrt{x} \). Shift every point on the graph 1 unit to the right and 2 units up. For example, the point (0,0) on \( y = \sqrt{x} \) moves to (1, 2) on \( y = \sqrt{x-1} + 2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

square root function
The square root function is a fundamental function in algebra. It is usually written as \( y = \sqrt{x} \). The graph of this function has several important characteristics
- It starts at the origin (0,0) and moves upwards and to the right.
- It is only defined for \( x \geq 0 \), as you cannot take the square root of a negative number in real numbers.
- It has a 'half-parabola' shape, curving gently upwards.
Understanding the basic properties and graph of the square root function helps in applying transformations such as shifts and stretches.
horizontal shift
A horizontal shift involves moving the graph of a function left or right on the coordinate plane. For the function \( y = \sqrt{x-1} \), the graph shifts 1 unit to the right. This is because we are subtracting 1 from the x-value inside the square root.
- When the equation reads \( y = \sqrt{x-a} \), where 'a' is a positive constant, the graph shifts 'a' units to the right.
- If it reads \( y = \sqrt{x+a} \), the graph shifts 'a' units to the left.
The direction of the shift is counterintuitive because subtracting from x shifts the graph to the right, while adding to x shifts it to the left.
vertical shift
A vertical shift moves the graph up or down along the y-axis. For the given function \( y = \sqrt{x-1} +2 \), it involves moving the graph 2 units upward. This shift is determined by the constant term outside the square root function.
- When the equation is \( y = \sqrt{x}+b \), where 'b' is a positive constant, the graph shifts 'b' units up.
- If it reads \( y = \sqrt{x}-b \), the graph moves 'b' units down.
This transformation is more straightforward compared to the horizontal shift as it directly adds or subtracts values to the y-coordinates of the points on the graph.

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