Problem 16 - Algebra and Trigonometry

Question

\(15-18\) Explain how the graph of \(g\) is obtained from the graph of \(f .\) $$ \begin{array}{l}{\text { (a) } f(x)=x^{3}, \quad g(x)=(x-4)^{3}} \\ {\text { (b) } f(x)=x^{3}, \quad g(x)=x^{3}-4}\end{array} $$

Step-by-Step Solution

Answer

(a) Right shift by 4; (b) Down shift by 4.

Step 1: Understand the Original Function

The original function is given as \(f(x) = x^3\). This is a cubic function that is symmetric about the origin, with an inflection point at \((0,0)\). It passes through the points \((-1,-1)\), \((0,0)\), and \((1,1)\), and has a typical 'S' shape.

Step 2: Transformation for Part (a) - Horizontal Shift

For \(g(x) = (x-4)^3\), the function \(g\) is derived from \(f\) by replacing \(x\) with \(x-4\). This indicates a horizontal shift of the graph of \(f(x)\) to the right by 4 units. Hence, every point \((x, y)\) on the graph of \(f(x)\) moves to \((x+4, y)\) on the graph of \(g(x)\).

Step 3: Transformation for Part (b) - Vertical Shift

For \(g(x) = x^3 - 4\), the function \(g\) is obtained by subtracting 4 from the function \(f(x) = x^3\). This results in a vertical shift of the graph of \(f(x)\) downward by 4 units. Therefore, every point \((x, y)\) on the graph of \(f(x)\) moves to \((x, y-4)\) on the graph of \(g(x)\).

Key Concepts

Cubic Function

The cubic function is one of the most basic forms of polynomial functions. It's defined as a function of the form \( f(x) = x^3 \). This mathematical expression produces an 'S'-shaped curve when graphed, known for its distinctive symmetry and unique properties.

• The cubic function is symmetric about the origin, meaning it looks the same when rotated 180 degrees around the origin.
• It has an inflection point at point (0,0), where the curve changes concavity.
• For the basic cubic function \( f(x) = x^3 \), it passes through points like \((-1, -1)\), \((0, 0)\), and \((1, 1)\).
• As \( x \) goes to positive or negative infinity, \( f(x) \) also goes to positive or negative infinity, respectively.

Understanding the nature of the cubic function is key when learning about function transformations, such as shifts and stretches, because each transformation modifies the original characteristics of the curve in specific ways. The transformations maintain the curve's overall shape but change its position or stretch it in the graph.

Horizontal Shift

The horizontal shift is a type of transformation that moves the position of a graph left or right in the coordinate plane. To create a horizontal shift, the input of the function is altered by adding or subtracting a constant. For example, when transforming the cubic function \( f(x) = x^3 \) to \( g(x) = (x-4)^3 \), you're performing a horizontal shift.

• In \( g(x) = (x-4)^3 \), every value of \( x \) is replaced by \( x-4 \), which shifts the entire graph to the right by 4 units.
• This shift affects the inflection point, moving it from (0,0) to (4,0), and all other points on the graph follow accordingly.
• A positive shift (using \( x-c \)) moves the graph to the right, while a negative shift (using \( x+c \)) would move it to the left.

Horizontal shifts do not change the shape or orientation of the graph, only its horizontal position.

Vertical Shift

A vertical shift moves the graph of a function up or down on the Cartesian plane. This transformation is achieved by adding or subtracting a constant from the function. For the cubic function in question,\( f(x) = x^3 \), transforming it to \( g(x) = x^3 - 4 \) results in a vertical shift.

• The vertical shift occurs when a constant is added to or subtracted from \( f(x) \). For \( g(x) = x^3 - 4 \), subtracting 4 shifts the entire graph downward by 4 units.
• This move impacts every point on the graph, including the inflection point, which moves from (0,0) to (0,-4).
• Additive transformations raise the graph, while subtractive transformations lower the graph.
• Vertical shifts also do not modify the shape or asymmetry of the graph; they solely affect its vertical position.

Horizontal and vertical shifts are basic yet pivotal transformations that facilitate creating new function graphs based on simple modifications of an existing graph.