Problem 73 - Algebra 2

Question

The graph of \(y=-\sqrt{x}\) is shifted 4 units up and 3 units right. Which equation represents the new graph? A. \(y=-\sqrt{x-4}+3\) B. \(y=-\sqrt{x-3}+4\) C. \(y=-\sqrt{x+3}+4\) D. \(y=-\sqrt{x+4}+3\)

Step-by-Step Solution

Answer

The correct equation that represents the new graph is \(y=-\sqrt{x-3}+4\), which corresponds to option B.

Step 1: Identify the Original Function

The original function is \(y=-\sqrt{x}\). This is a downward-opening square root function.

Step 2: Apply the Vertical Shift

The graph is shifted 4 units up. A 'units up' shift adds to the 'y' values. This transformed equation becomes \(y=-\sqrt{x} + 4\).

Step 3: Apply the Horizontal Shift

The graph is also shifted 3 units to the right. A 'units right' shift subtracts from the 'x' values. Therefore, the equation after this transformation becomes \(y=-\sqrt{x-3} + 4\).

Key Concepts

Vertical Shift

Vertical shifts occur when we adjust the graph of a function up or down on the Cartesian plane. In mathematical terms, this means adding or subtracting a constant to the entire function. For example, if we have a function such as \( f(x) = \sqrt{x} \). If we want to move this graph upwards by 4 units, we add 4 to the function: \[ g(x) = \sqrt{x} + 4 \].This transformation affects only the y-values, raising or lowering the graph without changing its shape.

• A positive constant shifts the graph upwards.
• A negative constant moves the graph downwards.

In our exercise, the graph of \( y=-\sqrt{x} \) is moved up by 4 units, resulting in the equation: \( y=-\sqrt{x} + 4 \). This addition makes it clear that each point on the graph is 4 units higher than its original position.

Horizontal Shift

Horizontal shifts change the position of a graph left or right. When graphing, this involves moving each point of the function horizontally across the x-axis. To achieve this shift in an equation, a constant is added or subtracted inside the function's argument.
A function, such as \( f(x) = \sqrt{x} \), can be shifted 3 units to the right by replacing \( x \) with \( x - 3 \) to form \( g(x) = \sqrt{x - 3} \).

• Subtracting a positive constant shifts the graph to the right.
• Adding a constant shifts the graph to the left.

This alteration directly modifies the x-values of each point. In the given task, shifting the graph of \( y=-\sqrt{x} \) 3 units right transforms the equation into \( y=-\sqrt{x-3} \), further altering the function to \( y=-\sqrt{x-3} + 4 \) when combined with the vertical shift.

Square Root Functions

Square root functions involve the square root of the variable, often producing a graph that begins at a particular point and extends rightward in a rising or falling curve. The parent function \( y=\sqrt{x} \) represents a basic upward-opening curve starting from the origin, and modifications to this can make the curve open downwards or move around.Key transformations concern direction and position:

• Negative signs in front, as in \( y=-\sqrt{x} \), invert the curve vertically, causing it to open downwards.
• Vertical and horizontal shifts move the graph up, down, left, or right without altering its directional characteristic.

The task at hand takes \( y=-\sqrt{x} \) and applies both vertical and horizontal shifts, demonstrating how graphs can be manipulated. By shifting the original downward-opening curve right and upwards, we transform it to \( y=-\sqrt{x-3} + 4 \). Understanding these basic transformations helps to grasp how complex graphs derive from simple root functions.