Question

Use a graphing utility. Graph \(y=x^{3}\). Then on the same screen graph \(y=(x-1)^{3}+2\). Could you have predicted the result?

Short answer

The graph is shifted 1 unit right and 2 units up.

Step-by-step solution

- Graph the Original Function

Use a graphing utility to plot the graph of the original function, which is given by the equation: y = x^3 Notice the curve and its symmetry about the origin.

- Graph the Translated Function

Next, on the same graph, plot the function y = (x - 1)^3 + 2. Observe how the new graph is a transformation of the original function y = x^3.

- Analyze the Transformation

Compare the two graphs. The second graph y = (x - 1)^3 + 2 is a translation of the first graph y = x^3. The term (x - 1) shifts the graph 1 unit to the right, and the +2 moves it 2 units up.

- Predicting the Result

By understanding translations in graphing functions, you could predict that the graph of y = (x - 1)^3 + 2 would be the graph of y = x^3 shifted 1 unit to the right and 2 units up.

Key concepts

Graphing utility

Graphing utilities are powerful tools that help visualize mathematical functions. These can be physical calculators with graphing capabilities or software on computers and mobile devices. Examples include Desmos, GeoGebra, and the graphing tools found in many scientific calculators. Using these tools, we can plot complex graphs quickly and accurately. To graph a function, you simply input the function equation, and the utility displays the graph. These tools also help in exploring various transformations and comparing different functions easily. For our exercise, you would input y = x^3 and y = (x-1)^3 + 2 into a graphing utility to see how the graphs relate.

Cubic functions

A cubic function is a type of polynomial function defined by the general equation y = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants, and a ≠ 0. The most basic form is y = x^3, which has some key characteristics:

• Its graph is a curve that passes through the origin (0,0).
• It has rotational symmetry about the origin.
• As x approaches infinity, y also approaches infinity; as x approaches negative infinity, y approaches negative infinity.

The cubic function y = x^3 is known for its unique 'S' shape and serves as a foundational example for understanding more complex transformations.

Function translation

Function translation involves shifting the graph of a function horizontally, vertically, or both. For a function y = f(x), a horizontal shift is achieved by modifying the x variable, such as y = f(x - h), where h is the horizontal shift. A vertical shift is done by adding or subtracting a constant k, resulting in y = f(x) + k. In our example function y = (x - 1)^3 + 2:

• (x - 1) moves the graph 1 unit to the right.
• The +2 raises the entire graph 2 units up.

By understanding these basic shifts, you can predict and sketch the new graph easily. This is a fundamental concept in graph transformations and crucial for analyzing various functions.