Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=\sqrt{x+3}-4$$

Step-by-Step Solution

Answer

Shift the graph of \(y = \sqrt{x}\) 3 units left and 4 units down.

Step 1: Identify the Base Function

In the given equation \(y = \sqrt{x+3} - 4\), the base function is \(y = \sqrt{x}\). This function forms the foundation of the graph we will create.

Step 2: Analyze the Horizontal Translation

The term \(x + 3\) inside the square root indicates a horizontal shift. Specifically, this translates the graph \(3\) units to the left because the form is \(\sqrt{x + h}\) where \(h = 3\), moving the function in the opposite direction.

Step 3: Analyze the Vertical Translation

The term \(-4\) outside the square root indicates a vertical shift. This translates the entire graph \(4\) units downward. This is consistent with the transformation \(y = f(x) - k\), where \(k = 4\).

Step 4: Sketch the Transformed Graph

Start by sketching the base graph \(y = \sqrt{x}\), which starts at the origin \((0,0)\) and curves upward to the right. Translate this entire graph \(3\) units to the left, making the starting point \((-3,0)\). Then translate the graph \(4\) units down, changing the starting point to \((-3,-4)\). Draw the transformed graph starting from this new origin and follow the same curve behavior.

Key Concepts

Understanding Base Functions

Base functions are the starting point for graphing equations. They are the simplest form of a function without any transformations applied. In the context of this exercise, the base function is \(y = \sqrt{x}\). This is a basic square root function that starts at the origin, denoted by the point \((0,0)\), and curves gently upwards to the right. This basic shape is used as the foundation on which shifts and other transformations are performed. Understanding the properties of these base functions is crucial, as it helps in predicting how the graph will change with various transformations.
Base functions can be other common mathematical functions such as:

• \(y = x^2\) - a parabola that opens upwards.
• \(y = x^3\) - a cubic function with distinct turning points.
• \(y = |x|\) - an absolute value function forming a V shape.

Each base function has its unique graph shape and helps determine the starting point for transformations like shifts, stretches, or reflections.

The Concept of Horizontal Shift

A horizontal shift moves the entire graph of a function to the left or right without altering its shape. In our example, the expression inside the square root, \(x+3\), indicates a horizontal shift. Generally, if you see something inside the function like \(f(x + c)\), it means that the graph will shift horizontally in the opposite direction of the sign. So, for \(\sqrt{x+3}\), the graph shifts left by 3 units.
With horizontal shifts:

• Addition inside a function (\(+ h\)) shifts the graph to the left by \(h\) units.
• Subtraction (\(- h\)) shifts it to the right by \(h\) units.

These transformations might initially seem counterintuitive because we usually associate positive changes with rightward movement. However, when it's inside the function, the effects are reversed.

Exploring Vertical Shift

Vertical shift is one of the most straightforward transformations. It involves moving the graph up or down along the y-axis. In the provided equation \(y = \sqrt{x+3} - 4\), the term \(-4\) outside the square root signifies a vertical shift downward by 4 units.
With vertical shifts:

• A negative number (\(- k\)) outside the function means moving the graph down by \(k\) units.
• A positive number (\(+ k\)) moves it up by \(k\) units.

Such shifts are intuitive because they affect the entire graph evenly, unlike horizontal shifts which require a change in the input variable \(x\). By understanding and identifying vertical shifts, you can easily find the new position of the graph after performing the vertical translation.