Problem 30 - Thomas Calculus

Question

Exercises \(25-34\) tell how many units and in what directions the graphs of the given equations are shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ y=-\sqrt{x} \quad \text { Right } 3 $$

Step-by-Step Solution

Answer

Shift right 3: \( y = -\sqrt{x-3} \).

Step 1: Understand the Original Graph

The given equation is \( y = -\sqrt{x} \). This equation represents a graph of a square root function that is reflected over the x-axis. The basic shape of the square root function starts from the origin and extends to the right, but due to the negative sign, it will open downwards.

Step 2: Identify the Direction and Units of Shift

The problem specifies a shift 'Right 3'. This means that every point on the graph will be moved 3 units to the right along the x-axis.

Step 3: Adjust the Equation for the Shift

To shift the graph of \( y = -\sqrt{x} \) to the right by 3 units, replace \( x \) with \( x - 3 \) in the equation. This results in the new equation: \( y = -\sqrt{x-3} \).

Step 4: Sketch the Original Graph

The original graph of \( y = -\sqrt{x} \) starts at the origin (0,0) and moves downwards and to the right. Plot several points such as (0,0), (1,-1), and (4,-2) to form the curve.

Step 5: Sketch the Shifted Graph

For the shifted graph \( y = -\sqrt{x-3} \), start at the new point (3,0). The graph will mimic the shape of the original but starts at (3,0) instead of (0,0). Plot points such as (3,0), (4,-1), and (7,-2) to visualize the shift.

Step 6: Label the Graphs

Clearly label the original graph with its equation \( y = -\sqrt{x} \) and the shifted graph with the equation \( y = -\sqrt{x-3} \) to distinguish between them.

Key Concepts

Square Root Function

A square root function is a type of algebraic function that involves the square root of a variable, often denoted as \( y = \sqrt{x} \). In its simplest form, the graph of a square root function starts at the origin point (0,0) and curves upward to the right. The shape is typically that of half a parabola lying on its side.

• Original form: \( y = \sqrt{x} \)
• Starts from the origin
• Curves to the right

However, when a negative sign is placed in front of the square root, as in the function \( y = -\sqrt{x} \), this reflects the graph over the x-axis. Now, instead of curving upwards, the graph will open downwards while still starting from the origin.
This basic understanding is crucial as it helps us recognize how shifts and other modifications will affect the graph.

Graph Shifting

Graph shifting involves moving every point of a graph a certain number of units in a specified direction without altering its shape. Shifting a graph horizontally means adjusting the graph left or right along the x-axis.

• Horizontal shifts: Move the graph along the x-axis.
• Vertical shifts: Move the graph along the y-axis.

In our specific problem, the task requires shifting the graph of \( y = -\sqrt{x} \) to the right by 3 units. Moving the graph to the right means adding a number within the square root function.
The shift is performed by replacing \( x \) with \( x-3 \) in the equation, resulting in \( y = -\sqrt{x-3} \). This transformation moves every point on the graph of the original function 3 units to the right while maintaining the same general shape and orientation.

Equation Modification

Modification of the equation is a key process in executing transformations like shifting. To shift a graph horizontally, we alter the variable within the function, usually by addition or subtraction. For example, to shift \( y = \sqrt{x} \) right by 3 units, we replace \( x \) with \( x-3 \).
For the problem at hand, the original equation \( y = -\sqrt{x} \) is transformed to \( y = -\sqrt{x-3} \) due to the rightward shift. The part \( x-3 \) indicates that every x-value of the original function has been increased by 3, forming the new position of the graph.
This modification demonstrates how the graph's layout on the coordinate plane is altered by the equation change. Key to these shifts is recognizing which part of the equation needs change.

• Right shift: Replace \( x \) with \( x-c \), where \( c \) is the number of shift units.
• Equation form directly affects graph movement.