Problem 46 - Algebra and Trigonometry

Question

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=x^{3} ;\) shift downward 1 unit

Step-by-Step Solution

Answer

The final transformed function is \( f(x) = x^3 - 1 \).

Step 1: Understand the Transformation

We start with the function \( f(x) = x^3 \). The transformation we need to apply is a vertical shift downward by 1 unit.

Step 2: Apply the Transformation

To shift the graph of a function downward by 1 unit, we subtract 1 from the function. This changes the function to \( f(x) - 1 \).

Step 3: Write the Transformed Function

After applying the transformation, the new function becomes \( f(x) = x^3 - 1 \). This represents the graph of the original function shifted down by 1 unit.

Key Concepts

Vertical Shift

In mathematics, a vertical shift is one of the simplest types of function transformations. It involves moving the entire graph of a function up or down without altering its shape.
When you perform a vertical shift, you directly add or subtract a constant from the function itself.
For example, if you have a function like a cubic function, let's say \( f(x) = x^3 \), and you want to apply a vertical shift:

• If you add a constant value, the function graph moves upwards. For example, \( f(x) + 1 \) shifts the graph up by 1 unit.
• If you subtract a constant value, the graph shifts downwards. For instance, \( f(x) - 1 \) shifts the graph down by 1 unit.

In our scenario, shifting \( f(x) = x^3 \) downward by 1 unit results in the function \( f(x) = x^3 - 1 \). This simple subtraction causes the entire cubic graph to move 1 unit lower on the cartesian plane.

Graph Transformations

Graph transformations allow us to manipulate the appearance and position of graphs in a plane.
These changes involve various operations such as translations, reflections, stretches, and compressions.

• Translations: These can be either vertical or horizontal movements. For example, the vertical shift we examined moves the graph up or down.
• Reflections: Flipping the graph over a specific line. For instance, reflecting over the x-axis changes \( f(x) \) to \( -f(x) \).
• Stretches/Compressions: These change the scale of the graph vertically or horizontally.

Before applying any transformation, it’s crucial to understand the original function's shape. For \( f(x) = x^3 \), you're starting with a cubic graph that looks like an elongated curve running from the lower left to the upper right. Transforming this graph involves shifts that keep its characteristics intact but change its location in the coordinate plane.

Cubic Functions

Cubic functions are polynomial functions of degree three and have the form \( f(x) = ax^3 + bx^2 + cx + d \).
They are known for certain unique features that distinguish them from linear and quadratic functions.

• Graph Shape: The graph of a cubic function usually resembles a soft 'S' curve. It can have one turning point and may intersect the x-axis at up to three points.
• Symmetry: If \( a = 1 \) and no other terms are present (e.g., \( f(x) = x^3 \)), it will exhibit point symmetry around the origin.
• End Behavior: As \( x \) approaches positive or negative infinity, the values of \( f(x) = x^3 \) rise or fall sharply.

In the specific case of \( f(x) = x^3 \), the function is very symmetric and lacks a constant term, so any shifts, such as the vertical shift to create \( f(x) = x^3 - 1 \), will simply move the entire soft 'S' curve downwards along the y-axis.