Problem 38 - Calculus with Concepts in Calculus

Question

Sketch the graph of the given equation with the help of a suitable translation. Show both the \(x\) and \(y\) axes and the \(X\) and \(Y\) axes. $$ y=\sqrt{x+3} $$

Step-by-Step Solution

Answer

The graph of \( y = \sqrt{x+3} \) is \( y = \sqrt{x} \) shifted 3 units left.

Step 1: Identify the Base Function

The given function is \( y = \sqrt{x+3} \). The base function of this equation is \( y = \sqrt{x} \), which is the graph of a square root function that starts at the origin (0,0) and increases to the right.

Step 2: Determine the Translation

In the equation \( y = \sqrt{x+3} \), the expression \( x+3 \) indicates a horizontal shift of the graph of \( y = \sqrt{x} \). Specifically, the graph is translated 3 units to the left on the x-axis, because the expression inside the square root is \( x+3 \), which shifts the graph opposite to the sign inside the root. Thus, \(x+3 = 0\) gives \(x = -3\).

Step 3: Sketch the Translated Graph

To sketch the graph, start with the base graph of \( y = \sqrt{x} \). Then shift every point on this graph 3 units to the left. Originally, the point (0, 0) is on the base graph, and after translation, it becomes (-3, 0). The shape of the graph remains the same as \( y = \sqrt{x} \) but starts at (-3, 0) and extends rightward.

Step 4: Label the Axes Clearly

Once the graph is sketched, label both the original x and y axes, as well as the translated X and Y axes. The translated X-axis is effectively the x-axis moved to coincide with the starting point of the graph at x = -3, representing the horizontal shift. The Y-axis remains the same in this case since there is no vertical translation involved.

Key Concepts

Square Root Functions

Square root functions are a type of radical function involving the root of a variable, typically shown as \( y = \sqrt{x} \). This is a special function characterized by its curved shape, which starts at a certain point and gradually rises to the right along the axis. For square root functions:

• The base function \( y = \sqrt{x} \) starts at the origin point \((0, 0)\), and increases smoothly to the right as \(x\) increases.
• The domain is all non-negative real numbers \( x \geq 0 \), since the square root of a negative number is not defined in the real number system.
• The range is also all non-negative real numbers \( y \geq 0 \), since the square root function produces non-negative outputs.

This fundamental understanding of the square root function is essential for comprehending transformations such as translations. By knowing what the starting graph looks like, we can more easily visualize and apply changes to it, such as shifts and stretches.

Horizontal Translation

Horizontal translation, also known as horizontal shift, refers to sliding the graph of a function left or right along the x-axis. This occurs without altering the shape or direction of the graph itself. For the function \( y = \sqrt{x+3} \):

• The term \( x+3 \) indicates a leftward shift of 3 units. Though it may seem counterintuitive, adding a positive constant causes the graph to move in the negative x-direction.
• Effectively, for the base function \( y = \sqrt{x} \), which starts at \((0, 0)\), the new start point for \( y = \sqrt{x+3} \) becomes \((-3, 0)\) after the translation.

To visualize this, imagine picking up the entire graph of \( y = \sqrt{x} \) and sliding it directly to the left by 3 units. The result is a graph that maintains its original arching shape, but now has its starting point at \( x = -3 \). Understanding this movement is key for accurately sketching translated graphs and predicting their behaviors.

Coordinate Systems

The concept of coordinate systems is foundational in graphing any function. Typically, we utilize the Cartesian coordinate system, which is set up using two perpendicular axes:

• The x-axis runs horizontally.
• The y-axis runs vertically.

In problems involving translations, it is important to distinguish between the original axes and any adjusted axes that represent the transformed graph. For example, when graphing \( y = \sqrt{x+3} \):

• Traditional x and y axes are used for the base function \( y = \sqrt{x} \).
• Once translated, the x-axis effectively becomes our newly imagined X-axis, aligned with where the new function graph begins at \( x = -3 \).
• The y-axis remains unchanged in this particular shift because the graph only moves horizontally.

Having clear labels for both sets of axes can help avoid confusion, especially in more complex problems where multiple transformations are applied. Always ensure that the graph and axes are labeled accurately to reflect these translations.