Problem 15 - Precalculus

Question

Sketch the graph of \(y=g(x)\) by starting with the graph of \(y=f(x)\) and using the transformations presented in Section 1.7 . $$f(x)=\sqrt[5]{x}, g(x)=\sqrt[5]{x+2}+3$$

Step-by-Step Solution

Answer

The graph of \( g(x) = \sqrt[5]{x+2}+3 \) is \( f(x) = \sqrt[5]{x} \) shifted 2 units left and 3 units up.

Step 1: Identify the Parent Function

The parent function provided is \( f(x) = \sqrt[5]{x} \). This is a transformed version of the basic power function \( y = x^{1/5} \), which passes through the origin and is symmetric about the origin.

Step 2: Determine Horizontal Shift Transformation

The expression \( g(x) = \sqrt[5]{x+2} + 3 \) indicates a horizontal shift. The term \( x+2 \) implies a shift of the graph 2 units to the left. The transformation rule applied is \( g(x) = f(x+2) \).

Step 3: Determine Vertical Shift Transformation

The \(+3\) outside the radical in \( g(x) = \sqrt[5]{x+2}+3 \) signifies a vertical shift. This moves the graph 3 units upward. The transformation rule applied here is \( g(x) = f(x) + 3 \).

Step 4: Combine Transformations

To obtain \( g(x) = \sqrt[5]{x+2}+3 \), we apply both transformations: first shift the parent function 2 units left, then 3 units up. Essentially, transform any point \((x, y) \) on \( f(x) \) to \((x-2, y+3)\) for \( g(x) \).

Step 5: Sketch the Graph

Start with the graph of \( f(x) = \sqrt[5]{x} \). Then apply the transformations, shifting every point left by 2 units and up by 3 units. This produces the graph of \( g(x) = \sqrt[5]{x+2}+3 \), with no change in shape or orientation.

Key Concepts

Parent Function

In math, it's crucial to understand what a parent function is because it serves as the foundation for various transformations. A parent function is the simplest form of a function in a family of functions. It's the original prototype and doesn't include any alterations such as shifts or stretches. For our exercise, the parent function is \( f(x) = \sqrt[5]{x} \), or equivalently, \( y = x^{1/5} \). This function is a type of power function that passes through the origin, where both the x and y values are zero. It’s a simple graph that helps us to comprehend how transformations impact the graph's shape and position. Understanding the parent function is vital because it allows us to predict how adding or subtracting values will modify the graph's appearance and location.

Horizontal Shift

Horizontal shifts involve moving the graph of a function left or right. They occur whenever there is a transformation inside the function's input (or the variable, typically represented as \( x \)). In our equation, \( g(x) = \sqrt[5]{x+2} + 3 \), the term \( x+2 \) indicates a horizontal shift. Specifically, it means the graph has shifted 2 units to the left.

• When the function appears as \( f(x+a) \), the graph shifts left by \( a \) units.
• When it is \( f(x-a) \), the graph moves right by \( a \) units.

For horizontal shifts, it's essential to understand that adding a number inside the function causes the graph to move in the opposite direction to what might be intuitively expected. That means \( x+2 \) moves left, not right. By grasping this concept, predicting the position of a transformed graph becomes much easier.

Vertical Shift

Vertical shifts involve moving the graph up or down along the y-axis. In the expression \( g(x) = \sqrt[5]{x+2} + 3 \), the \(+3\) component tells us that the function's graph has moved up three units. This type of transformation doesn't affect the shape or orientation of the graph, only its vertical position.

• a positive value such as \( +3 \) in \( f(x) + 3 \) shifts the graph upwards.
• a negative value like \( -3 \) would shift it downwards.

Like horizontal shifts, vertical shifts are simple to perform: you merely add or subtract a constant from the entire function. This transformation is straightforward because it doesn't change the function's original structure, only its location on the y-axis.

Power Function

A power function is any function that can be expressed in the form \( y = x^n \), where \( n \) is a constant. These functions are very versatile and can take many shapes depending on the exponent. For example:

• If \( n = 1 \), the function is a straight line, \( y = x \).
• When \( n = 2 \), it's a parabola, \( y = x^2 \).
• Our function has \( n = 1/5 \), creating a curve that gets less steep as it moves away from the origin.

In the solution, the parent function \( y = x^{1/5} \) is a specific type of power function. It defines a characteristic shape that changes through transformations like shifts but maintains its core nature. Knowing the base shape and properties of a power function enables one to skillfully apply changes and predict the new graph's appearance.