Problem 24 - Thomas Calculus

Question

Exercises \(19-28\) tell how many units and in what directions the graphs of the given equations are to be 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 3 units right: New equation is \(y = -\sqrt{x-3}\).

Step 1: Identify the Original Function

The original equation given is \(y = -\sqrt{x}\). This is the equation of a graph where the negative square root of \(x\) is plotted. It is a transformed version of the basic square root function \(y = \sqrt{x}\) reflected across the x-axis.

Step 2: Determine the Shift

The problem states that the graph of the function should be shifted 'Right 3'. This indicates a horizontal shift of the graph 3 units to the right on the Cartesian plane.

Step 3: Formulate the Shifted Equation

To shift a function horizontally by \(c\) units, you modify the input \(x\) with an opposite sign. Therefore, shifting \(y = -\sqrt{x}\) 3 units to the right will result in the equation \(y = -\sqrt{x-3}\).

Step 4: Sketch the Graphs

First, sketch the graph of the original function \(y = -\sqrt{x}\). This graph should start at the origin, moving downwards and to the right. Then, for the shifted function \(y = -\sqrt{x-3}\), move every point on the original graph 3 units to the right. Label each graph accordingly.

Key Concepts

Horizontal Shift

A horizontal shift involves moving the entire graph of a function left or right on the Cartesian plane. The direction and number of units to move are crucial for this transformation.

• A shift to the right is achieved by replacing the variable \(x\) with \(x - c\), where \(c\) is the number of units.
• A shift to the left is achieved by replacing \(x\) with \(x + c\).

In the given exercise, the original function \(y = -\sqrt{x}\) is shifted to the right by 3 units. This means you adjust the input, \(x\), so the function becomes \(y = -\sqrt{x-3}\). Every point on the original graph is moved 3 units to the right, maintaining the same shape but in a new location.

Vertical Shift

A vertical shift modifies the position of a graph up or down along the y-axis. The vertical shift depends on whether you are adding or subtracting from the entire function.

• To shift upwards, add a constant \(k\) to the function, creating the new function \(y = f(x) + k\).
• To shift downwards, subtract \(k\) from the function, resulting in \(y = f(x) - k\).

In our particular problem, a vertical shift is not applied, but understanding how it works is key for similar problems. Adding or subtracting from the entire function affects the position along the y-axis, leaving the x-coordinates unchanged.

Reflected Graph

Reflection in graphs involves flipping the graph across a particular axis, changing how it is presented visually.

• Reflecting across the x-axis is done by changing the function from \(y = f(x)\) to \(y = -f(x)\).
• Reflecting across the y-axis involves substituting \(x\) with \(-x\), i.e., from \(y = f(x)\) to \(y = f(-x)\).

In the original problem, the function \(y = -\sqrt{x}\) involves a reflection of the basic square root function \(y = \sqrt{x}\) across the x-axis. This reflection creates a graph that appears upside-down, resulting in a downward trajectory from the origin to the right, illustrating the negative impact on the values of \(y\).

Square Root Function

The square root function is foundational and appears in many mathematical contexts. The basic form is \(y = \sqrt{x}\), representing a half-parabola opening to the right. It starts at the origin (0,0) and moves upwards to the right as \(x\) increases.Significant characteristics of a square root function include:

• Domain: Non-negative values of \(x\), since square roots of negative numbers are not real.
• Range: Non-negative values of \(y\) for \(y = \sqrt{x}\) but changes with different transformations.
• Graph Shape: Starts at one point and moves in a positive or negative direction based on transformations.

For the exercise, we start with \(y = -\sqrt{x}\), transforming the basic square root graph into a reflected version, meaning it exists only in the negative part of the y-axis, changing its domain and range influence.