Problem 12 - College Algebra and Calculus: An Applied Approach

Question

Use the graph of \(y=x^{3}\) to sketch the graph of the function. $$f(x)=-(x-2)^{3}+2$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=-(x-2)^{3}+2\) is a rightward and upward shift of the graph of the base function \(y=x^{3}\), also reflected across the x-axis.

Step 1: Analyze the base function

The base function is \(y=x^{3}\), which is a cubic function. The graph of this function exhibits a distinct 's' shape, passing through the origin \( (0,0) \). It increases for negative x-values, decreases at x=0, and then increases again for positive x-values.

Step 2: Recognizing the transformations

The function \(f(x)=-(x-2)^{3}+2\) incorporates two transformations of the base function: a reflection and a shift. The negative sign in front of the cubic function indicates a reflection of the graph in the x-axis. The '-2' inside the function and the '+2' outside it represent a rightward shift by 2 units and an upward shift by 2 units, respectively.

Step 3: Apply the transformations

Start by reflecting the base function across the x-axis. Next, shift the graph right by 2 units and up by 2 units. This produces the final graph of the function \(f(x)=-(x-2)^{3}+2\).

Key Concepts

cubic function

A cubic function is an algebraic function of the form \(y = x^3\). When graphed, it typically exhibits a distinctive 'S' shape. This means it starts off increasing, then dips around the origin, before rising again. This 'S' curve crosses the origin point (0,0) and continues infinitely in both directions along the x-axis. The core characteristic of cubic functions is their power of three, which dictates both the curve's shape and behavior. Each point on the graph of a cubic function has a specific x-value, and there is one y-value corresponding to this x-value within the range of real numbers. The symmetry around the origin means negative x-values mirror positive x-values.

reflection across x-axis

Reflection across the x-axis is a transformation that flips the graph of a function over this axis. Mathematically, if you have a function \(y = f(x)\), its reflection would be \(y = -f(x)\).

For functions like the cubic function \(y = x^3\), applying this reflection changes how the s-shaped curve behaves. Instead of rising to the top right, dipping at the origin, and rising to the top left, the curve is inverted.

• The direction of increase and decrease is altered.
• The graph's 'shape' is maintained but its orientation is mirrored vertically.

This reflection affects all points: positive y-values become negative and vice versa. The graph appears to be upside down, maintaining symmetry concerning the x-axis.

graph shifting

Graph shifting is another transformative operation involving moving the entire graph of a function horizontally, vertically, or both.

For the function \(f(x) = -(x - 2)^3 + 2\):

• The term \(x - 2\) indicates a horizontal shift to the right by 2 units, since the subtraction of 2 "moves" the cubic shape along the x-axis.
• The \(+2\) outside of the function manifests as a vertical shift upwards by 2 units, lifting the entire graph along the y-axis.

Together, these shifts modify the original graph’s location but don't alter its shape. This procedure helps position the graph correctly in context, reflecting real-world movements like changes in equations based on different starting points or vertical displacements.

algebraic functions

Algebraic functions include a variety of operations and transformations applied to variables and constants within an expression. These can involve addition, subtraction, multiplication, division, and the raising of powers, as seen in polynomial functions like cubic ones.

One of the fundamental aspects of algebraic functions is their ability to transform and model real-world phenomena using unique properties of each function type. They can include simple linear relationships or more complex cubic expressions. \(f(x) = -(x - 2)^3 + 2\) is an example of a transformed algebraic function, taking characteristics of its base and reflecting them through shifts and inversions to form a new function. These transformations illustrate how we can manipulate equations creatively to understand and visualize scenarios through graphical representation.