Problem 45 - A Graphical Approach to College Algebra

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. $$y=x^{3}+1$$

Step-by-Step Solution

Answer

Shift the graph of \( y = x^3 \) up 1 unit for \( y = x^3 + 1 \).

Step 1: Identify the Base Function

The given function is \( y = x^3 + 1 \). The base function here is \( y = x^3 \), which is a cubic function.

Step 2: Understand the Transformation

The transformation of \( y = x^3 \) to \( y = x^3 + 1 \) involves a vertical shift. The constant \(+1\) indicates that the entire graph of the base function \( y = x^3 \) is shifted upwards by 1 unit.

Step 3: Draw the Base Function

Sketch the graph of the base function \( y = x^3 \). This function is shaped like an elongated 'S' curve, passing through the origin (0,0), and is symmetric about the origin.

Step 4: Apply the Vertical Shift

Shift the graph of \( y = x^3 \) upward by 1 unit to account for the \(+1\) in \( y = x^3 + 1 \). This means every point on \( y = x^3 \) will have its y-coordinate increased by 1. For example, the origin (0,0) moves to (0,1).

Step 5: Sketch the Final Graph

Draw the graph of \( y = x^3 + 1 \) on the same coordinate system. Ensure that the shape remains consistent with \( y = x^3 \) but is simply shifted upwards, so the curve will now pass through points like (0,1) instead of (0,0).

Key Concepts

Cubic Function

Let's start by looking at the cubic function, which is defined as \(y = x^3\). This function forms a curve that looks like an elongated 'S.' It is continuous and has a point of origin at (0,0).
A few characteristics of the cubic function include:

• It is symmetric about the origin, meaning if you rotate the graph 180 degrees around the origin, it looks the same.
• As \(x\) becomes very large positively or negatively, \(y\) will also increase or decrease correspondingly.
• The cubic function crosses through the origin (0,0).

These features make the cubic function quite distinct among other polynomial functions. Understanding this shape is crucial for graphing its transformations, like vertical shifts.

Vertical Shift

Now let's dive into the concept of a vertical shift in a graph. A vertical shift implies moving a graph up or down along the y-axis. When you add or subtract a constant to a function's output, you're performing a vertical shift.
In our exercise, we are transforming \(y = x^3\) into \(y = x^3 + 1\). This means we have a vertical shift of +1 unit.
Such transformation means every point on the original graph moves up by this constant:

• (0,0) in \(y = x^3\) moves to (0,1) in \(y = x^3 + 1\).
• All other points follow suit, maintaining the shape but shifted vertically.

This shift hasn't changed the shape of the graph, only its position on the y-axis.

Graph Sketching

Graph sketching is an essential skill. The goal is to draw a rough outline of the function without needing precise calculations. By sketching, you visualize how algebra translates to geometry.
For our function \(y = x^3 + 1\), we begin by sketching \(y = x^3\), which is already known for its 'S' shape through the origin. Then, apply the vertical shift, lifting the 'S' shape so it passes through (0,1).
When sketching, remember:

• Identify the base function: Know the basic shape.
• Apply transformations carefully: Move each point according to the operation.
• Maintain symmetry: The shape should retain its inherent characteristics, like symmetry.
• Use a smooth curve: The cubic function doesn't have sharp turns.

With practice, sketching becomes more intuitive and effective for understanding transformations.

Basic Functions

Basic functions like \(y=x^3\), \(y=x^2\), \(y=\sqrt{x}\), and \(y=|x|\) serve as building blocks for more complex graphs. They each have unique shapes and characteristics that form the foundation of graph transformations.
Understanding these basic functions allows you to:

• Predict the transformations' effects more easily.
• Recognize transformations visually on a graph.
• Analyze how different constants modify these standard shapes.

For instance, the cubic function \(y = x^3\) displays rotational symmetry, while the quadratic \(y = x^2\) is parabolic and symmetric along the y-axis. Knowing these subtle differences helps in accurately sketching and transforming graphs using simple operations like vertical shifts.