Problem 61

Question

Exer. 59-62: Sketch the graph of the equation. $$ y=|\sqrt{x}-1| $$

Step-by-Step Solution

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Answer

Graph reflects upward from (0, 1), passes through (1, 0), and follows $ \sqrt{x} - 1 $ for $ x > 1 $.

1Step 1: Understand the Components

The equation is given as $ y = |\sqrt{x} - 1| $. We need to understand that $ |z| $ denotes the absolute value function. This means it takes whatever is inside $ |z| $ and makes sure it is non-negative.

2Step 2: Analyze the Function $ \sqrt{x} - 1 $

First, let's consider $ y = \sqrt{x} - 1 $. The function $ \sqrt{x} $ is defined for $ x \geq 0 $ and starts at $ (0, -1) $ when $ x = 0 $. As $ x $ increases, $ \sqrt{x} $ becomes larger.

3Step 3: Apply Absolute Value

Since we have an absolute value, $ y = |\sqrt{x} - 1| $, the negative values of $ \sqrt{x} - 1 $ will be flipped to positive. This means when $ \sqrt{x} - 1 < 0 $, the graph will reflect above the x-axis.

4Step 4: Determine Key Points

Find when $ \sqrt{x} - 1 = 0 $. Solving gives $ \sqrt{x} = 1 $ or $ x = 1 $. Before $ x = 1 $, the graph reflects off the x-axis and after $ x = 1 $, it follows the line $ \sqrt{x} - 1 $.

5Step 5: Sketch the Graph

The graph of $ y = |\sqrt{x} - 1| $ starts at $ (0, 1) $, moving to $ (1, 0) $. For $ x > 1 $, the line follows $ y = \sqrt{x} - 1 $ upwards. Reflect the segment from $ x = 0 $ to $ x = 1 $ above the x-axis.

Key Concepts

Graph SketchingSquare Root FunctionFunction Transformation

Graph Sketching

Graph sketching involves plotting the curve of an equation to understand its behavior over different values of x. To sketch the graph of the function given by $ y = |\sqrt{x} - 1| $, start by identifying the main components of the equation. Here, the absolute value and square root play key roles. Start by considering the inner part, $ \sqrt{x} - 1 $, alone. This will help in understanding where the graph will be reflected.

Plot the point $ (0, -1) $ on the graph because $ \sqrt{0} - 1 = -1 $.
The graph rises as $ x $ increases, reaching $ (1, 0) $ when $ \sqrt{x} = 1 $.

Now, apply the absolute value, which reflects negative values to positive values. For values where $ \sqrt{x} - 1 < 0 $, reflect those points above the x-axis.

This creates a new starting point at $ (0, 1) $ instead of $ (0, -1) $.
Join the points smoothly, and for $ x > 1 $, continue the upward path along $ y = \sqrt{x} - 1 $.

Finally, the graph consists of two segments: one reflected above the x-axis from $ x = 0 $ to $ x = 1 $, and the other rising linearly for $ x > 1 $. This visual representation is essential to fully grasp the function's transformation.

Square Root Function

The square root function, $ \sqrt{x} $, is a fundamental concept in mathematics that forms the basis of our exercise. It's defined for $ x \geq 0 $ as real numbers and graphically represented as a curve starting at the origin and gradually increasing. The shape of $ \sqrt{x} $ is a smooth curve that rises only in the positive direction of the y-axis.

At $ x = 0 $, the value of $ \sqrt{x} $ is $ 0 $, so the point is $ (0, 0) $.
As $ x $ increases to $ 1 $, $ \sqrt{x} = 1 $, moving the point to $ (1, 1) $.

This intuitive rise means that the slope of the function decreases as $ x $ gets larger. In our exercise, $ \sqrt{x} $ is shifted downwards by 1, creating the expression $ \sqrt{x} - 1 $.

Every value of $ \sqrt{x} $ gets reduced by 1, moving $ (0, 0) $ to $ (0, -1) $ and $ (1, 1) $ to $ (1, 0) $.

Understanding the nature of the square root function helps in seeing the transformation and positioning each point accurately on your graph.

Function Transformation

Function transformation helps us manipulate basic functions to produce new curves. With transformations, you can shift, stretch, or reflect functions. Our focus in the exercise is mainly shifting and reflecting with the absolute value transformation.

A downward shift by 1 occurred through the term $ -1 $ in $ \sqrt{x} - 1 $.
When adding the absolute value, $ y = |\sqrt{x} - 1| $, the graph reflects parts below the x-axis to above.

When addressing transformations, it is essential to understand how operations change the function. A vertical translation simply moves the graph upwards or downwards, altering the y-values. Reflecting by an absolute value function impacts the symmetry, flipping below-zero values to ensure the output remains non-negative.

The graph of $ y = |\sqrt{x} - 1| $ starts reflecting from $ (0, -1) $ to $ (0, 1) $.
This transformation gives it a distinct V-like shape up to $ x = 1 $, after which it returns to its original path.

These transformations result in the shape and direction changes that are critical for graph sketching effectively.

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