Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions. \(k(x)=(x-2)^{3}-1\)

Step-by-Step Solution

Answer

Shift the cubic graph right 2 units and down 1 unit.

Step 1: Identify the Parent Function

The parent function for the given function is the cubic function, which is expressed as \( f(x) = x^3 \). This function features a graph that starts from the bottom left and moves to the top right, passing smoothly through the origin.

Step 2: Determine Horizontal Shifts

The function \( k(x) = (x-2)^3 - 1 \) involves a transformation of \( x-2 \). This indicates a horizontal shift of the graph. Specifically, the graph shifts 2 units to the right.

Step 3: Determine Vertical Shifts

The \(-1\) at the end of the function \( k(x) = (x-2)^3 - 1 \) is a vertical shift. This means the graph moves down by 1 unit from its position after the horizontal shift.

Step 4: Sketch the Transformed Graph

To sketch the transformed function \( k(x) = (x-2)^3 -1 \), start by drawing the cubic graph of the parent function \( f(x) = x^3 \). Shift this graph 2 units to the right, and then 1 unit down. The resulting curve should maintain the shape of the cubic function, passing through the point (2,-1).

Key Concepts

Parent Function

In the world of graph transformations, the **parent function** is your starting point. It's the most basic form of a function, like a template that other functions are based on. For our exercise, the parent function is the cubic function, noted as \( f(x) = x^3 \). The graph of a cubic function typically has an S-shape. It starts from the bottom left of the graph, passes smoothly through the origin (which is where the x-axis and y-axis meet), and continues to the top right. Knowing the shape and behavior of your parent function makes it easier to apply transformations and predict how the graph will look after changes.

Horizontal Shift

A **horizontal shift** in graph transformations means moving the graph left or right. This occurs when a constant is added or subtracted inside the function's argument. For the function \( k(x) = (x-2)^3 - 1 \), the `(x-2)` part suggests a horizontal shift. The graph of the parent function \( f(x) = x^3 \) is translated to the right by 2 units. Remember, if the function changes in the form of \( (x - h)^n \), the shift is to the right by \( h \), and if it changes to \( (x + h)^n \), the shift is to the left by \( h \). This helps adjust the graph horizontally without altering its shape.

Vertical Shift

In transformations, a **vertical shift** moves the graph up or down along the y-axis. This is reflected by adding or subtracting a constant outside the function. In our function \( k(x) = (x-2)^3 - 1 \), the \'-1\' indicates a vertical shift downwards by 1 unit. Opposite to horizontal shifts, the +/- sign describes exactly the direction of the movement. Whenever you see \( + c \) outside the function, move the graph up; if it’s \( - c \), the graph shifts down. Vertical shifts are straightforward and don't impact the symmetry or curvature of the graph. They merely adjust its position relative to the y-axis.

Cubic Function

The **cubic function** is a star among polynomial functions, exemplified by \( f(x) = x^3 \). It's a non-linear function with a unique S-shaped curve. This curve gradually increases on both sides of its inflection point, which is the origin (0,0). This function continues to rise and fall without bounds as x values grow towards positive and negative infinity. Cubic functions are symmetric around the origin, which means flipping the graph upside down or reflecting it across the y-axis produces the same graph. Understanding the basic shape and properties of the cubic function allows you to better predict how it will look after shifts and transformations.