Problem 54 - 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. $$y=(x+3)^{3}-1$$

Step-by-Step Solution

Answer

Shift \(y=x^3\) left 3 units and down 1 unit; new vertex is at \((-3, -1)\).

Step 1: Identify the base function

The given function is \( y = (x+3)^3 - 1 \). The base function here is \( y = x^3 \). This indicates the initial shape of our graph will be a cubic curve.

Step 2: Analyze the transformations

Start by analyzing the expression \( (x+3)^3 \). The \(x+3\) inside the parentheses indicates a horizontal shift. Specifically, this is a leftward shift by 3 units from the base function \( y = x^3 \). The \(-1\) outside the parentheses indicates a vertical shift, moving the graph downwards by 1 unit.

Step 3: Determine the new vertex

The vertex (or central turning point) of the base function \( y = x^3 \) is at \((0, 0)\). Due to the transformations, the new vertex after the shifts is \((-3, -1)\). This shifts the starting point of the cubic curve to reflect these changes.

Step 4: Sketch the graph

Start by placing the new vertex of the graph at \((-3, -1)\). The shape of the graph will have the typical "S" shape of a cubic function, with the central point passing through the vertex \((-3, -1)\). Make sure to maintain the direction and steepness of the cubic function as it extends both upward to the right and downward to the left.

Key Concepts

Cubic Functions

Cubic functions are a fascinating family of polynomial functions that have the general form of \( y = ax^3 + bx^2 + cx + d \). These functions often create a distinctive "S" shaped curve, referred to as a cubic curve.

• The cubic function is one whose highest power of the variable is 3.
• Its graph usually crosses the x-axis at most three times, giving up to three real roots.
• This type of function will have one inflection point where the curvature changes direction, known as the point of symmetry.

An example of a simple cubic function is \( y = x^3 \), which passes through the origin and symmetrically balances around it. Cubic functions can be stretched, compressed, or shifted, but they always maintain their characteristic "S" shape.

Graph Sketching

Graph sketching involves plotting the key features of a mathematical function without the use of digital tools.

• Start by determining the base shape of the function, for example, a cubic in the form \( y = x^3 \).
• Identify notable changes such as shifts and stretches.
• Plot key points, like intercepts and the vertex for cubic curves, then extend the graph to other areas, maintaining its natural flow.

In this scenario, you will begin at the new vertex location due to shifts before progressing to sketch the rest of the graph. This makes the process of drawing more logical and visually accurate.

Horizontal and Vertical Shifts

Function transformations such as horizontal and vertical shifts can significantly alter the appearance of a graph.
A horizontal shift moves the graph left or right in the x-direction. In the equation \( y = (x+3)^3 \), adding 3 inside the function's bracket moves it three units to the left. Basically, you replace \(x\) with \(x+c\) to move left, and \(x-c\) to move right.

• Horizontal shifts: Change inside the brackets of the function, affecting the position along the x-axis.
• Vertical shifts: Modify the function outside the main term, moving the graph up or down along the y-axis.

With the given function \( y = (x+3)^3 - 1 \), we note this displays both a leftward (horizontal) shift and downward (vertical) shift. The subtraction of 1 outside the bracket decreases the entire function value, shifting the graph one unit down. Understanding and plotting these transformations help in accurately sketching and analyzing graph behaviors.