Problem 131 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) $$g(x)=(x+1)^{3}-3$$

Step-by-Step Solution

Answer

The function \(g(x) = (x+1)^{3} - 3\) is a transformation of the function \(f(x) = x^{3}\), resulting from a horizontal shift one unit to the left and a vertical shift three units down.

Step 1: Identify the Functions

The given functions are \(f(x) = x^{3}\) and \(g(x) = (x+1)^{3} - 3\)

Step 2: Sketch the Graph of \(f(x) = x^{3}\)

The function \(f(x) = x^{3}\) is a cubic function that is symmetric with respect to the y-axis. The graph starts at negative infinity on the left, crosses through the origin (0,0), and extends to positive infinity on the right.

Step 3: Sketch the Graph of \(g(x) = (x+1)^{3} - 3\)

The function \(g(x) = (x+1)^{3} - 3\) is a transformation of the function \(f(x) = x^{3}\). Specifically, the \( +1\) inside the parentheses results in a shift to the left by one unit, and the \( -3\) at the end translates the graph down by three units. So, the graph of \(g\) will also have a symmetric shape with respect to the y-axis but will be shifted one unit left and three units down compared to the graph of \(f\). The graph of \(g\) starts at negative infinity on the left, crosses through the point (-1,-3), and extends to positive infinity to the right.

Step 4: Describe The Transformation

The transformation from function \(f\) to function \(g\) can be described as a horizontal shift of one unit to the left followed by a vertical shift of three units downward.

Key Concepts

Cubic Functions

Cubic functions are a special type of polynomial function. They are of the form \(f(x) = ax^3 + bx^2 + cx + d\) where \(a\) is not equal to zero. In the simplest case, which is \(f(x) = x^3\), the graph is symmetric around the origin. What makes cubic functions interesting is their distinctive 'S' shape.
These functions have both negative and positive sections. As you extend to the left of the graph, the function dives toward negative infinity. Conversely, as you move to the right, it climbs toward positive infinity. These graphs usually intersect the x-axis at one or more points, and the nature of these intersections can tell us about the roots of the cubic function.
Understanding the basic shape of a cubic function is essential for graph transformations.

Horizontal Shift

Graph transformations often involve shifting the graph horizontally without altering its shape. For the function \(g(x) = (x + 1)^3 - 3\), the expression \((x + 1)\) indicates a horizontal shift.
A horizontal shift means you move the entire graph left or right along the x-axis. For transformations like this, the shift depends on the "inside" function, \((x + c)\). If \(c\) is positive, the shift is actually to the left. If \(c\) were negative, it would make the shift rightward.

• \((x + 1)^3\): denotes a shift one unit to the left, because of the +1 inside the parentheses.

This action does not change the basic shape or orientation of the graph. Every point, including all significant features like vertices and intercepts, moves consistently in the horizontal direction.

Vertical Shift

In the context of transformations, a vertical shift involves moving the entire graph up or down on the y-axis. For the function \(g(x) = (x + 1)^3 - 3\), the term \(-3\) indicates such a shift.
Vertical shifts are controlled by adding or subtracting a constant outside of the function's main expression. If you subtract a positive number, the shift is downward, while adding results in an upward movement.

• \(-3\) in \((x+1)^3 - 3\): moves the graph 3 units down.

This transformation does not affect the horizontal alignment of the graph. It consistently lowers every point on the graph, impacting its vertical position without changing its horizontal position. Both horizontal and vertical shifts are vital for understanding how graphs are re-positioned without changing their fundamental shape.