Problem 8 - Algebra 2

Question

Graph each function. \(y=\sqrt{x-4}\)

Step-by-Step Solution

Answer

The graph of the given function \(y = \sqrt{x-4}\) starts at the point (4,0) and increases towards positive infinity. It is situated entirely within the first quadrant of the Cartesian plane.

Step 1: Identify the parent function and its basic shape

The parent function for our given function is \(y = \sqrt{x}\). This is a standard square root function, and its basic shape is a curve that starts from the origin (0,0) and goes towards positive infinity. It only exists in the first quadrant of the Cartesian plane.

Step 2: Ascertain the shift which is implied by the function

The given function is \(y = \sqrt{x-4}\). The '-4' inside the square root implies a horizontal shift to the right by four units. This means that instead of our curve starting at the origin, it will start at (4,0).

Step 3: Sketch the graph

Begin with an x-y axis, and plot the starting point of the curve at (4,0). From here, sketch the curve in an increasing manner towards positive infinity. The curve should only exist in the first quadrant (as with the parent function) and it will begin at x=4 instead of the origin.

Key Concepts

Horizontal Shift

When learning about graphing square root functions, a horizontal shift is an essential concept to understand. A horizontal shift occurs when a constant is added or subtracted from the variable inside the square root, altering the position of the graph on the x-axis.
In the function given, \(y = \sqrt{x-4}\), the number '-4' represents the horizontal shift. This value inside the function translates the graph. Specifically, the graph of the function moves to the right by 4 units on the Cartesian plane.

• This change impacts where the graph starts on the x-axis. Instead of beginning at \((0,0)\), it now starts at \((4,0)\).
• The rest of the graph follows the same shape as the parent function but shifts entirely to the right.

Understanding this concept is crucial for accurately sketching and analyzing such graphs.

Parent Function

A parent function is the simplest form of a function family, which acts as the foundation for other functions. For the function \(y = \sqrt{x-4}\), the parent function is \(y = \sqrt{x}\).
This parent function, \(y = \sqrt{x}\), creates a specific curve starting at the origin, \((0,0)\), on the Cartesian plane.

• It produces an increasing curve only existing in the first quadrant, indicating it only includes positive x and y values.
• The function's basic shape begins at a single point and steadily moves towards positive infinity.

Understanding the parent function is key as it helps you recognize how transformations, such as shifts or reflections, alter the graph's basic properties.

Cartesian Plane

The Cartesian plane is an essential tool in graphing functions, consisting of two perpendicular axes that intersect at a point called the origin. The horizontal axis is the x-axis, while the vertical axis is the y-axis.
When graphing functions like \(y = \sqrt{x-4}\), the Cartesian plane allows you to visually represent changes, such as the horizontal shift.

• The plane is divided into four quadrants, with the function \(y = \sqrt{x}\) typically residing in the first quadrant.
• This quadrantal positioning means it deals with only positive values for both x and y.

Therefore, the Cartesian plane provides a structured way to see how various functions compare to their parent functions and how shifts affect their location and orientation on the graph.