Question

a) Predict the relationship between the graphs of \(y=x^{3}-x^{2}\) and \(y=(x-2)^{3}-(x-2)^{2}\). b) Graph each function using technology to verify your prediction. c) Factor each function in part a) to determine the \(x\) -intercepts.

Short answer

The graph of \(y = (x-2)^3 - (x-2)^2\) is a 2-unit right translation of \(y = x^3 - x^2\). The x-intercepts are \(x=0\), \(x=1\), \(x=2\), and \(x=3\).

Step-by-step solution

- Understanding the Functions

We have two functions: 1) \(y = x^3 - x^2\) and 2) \(y = (x-2)^3 - (x-2)^2\). The second function is a transformation of the first function.

- Predicting the Relationship

Notice that \(y = (x-2)^3 - (x-2)^2\) is a horizontal translation of \(y = x^3 - x^2\) by 2 units to the right. This means the graph of the second function should look identical to the graph of the first function, but shifted 2 units to the right.

- Graphing the Functions

Use technology (like a graphing calculator or software) to plot both functions. Verify that the graph of \(y = (x-2)^3 - (x-2)^2\) is indeed the same as the graph of \(y = x^3 - x^2\) but shifted 2 units to the right.

- Factoring \(y = x^3 - x^2\)

\(y = x^3 - x^2 = x^2(x - 1)\). The factored form gives us the x-intercepts at \(x=0\) and \(x=1\).

- Factoring \(y = (x-2)^3 - (x-2)^2\)

\(y = (x-2)^3 - (x-2)^2 = (x-2)^2((x-2) - 1) = (x-2)^2(x-3)\). The factored form gives us the x-intercepts at \(x=2\) and \(x=3\).

Key concepts

Function Transformation

Understanding function transformation is crucial in understanding how changes in a function affect its graph. The exercise above involves two functions, namely, \( y = x^3 - x^2 \) and \( y = (x-2)^3 - (x-2)^2 \). The second function is a transformed version of the first one because we replaced \( x \) with \( x-2 \).
This is known as a horizontal translation. When we translate a function horizontally, we move its graph left or right. For example, adding 2 inside the function as \( (x-2) \) shifts the graph 2 units to the right. In our case, the entire graph of \( y = x^3 - x^2 \) shifts 2 units to the right to become \( y = (x-2)^3 - (x-2)^2 \).
This transformation means every point on the graph of \( y = x^3 - x^2 \) moves 2 units to the right. You can visualize this transformation by plotting both functions and observing the shift.

Factoring Polynomials

Factoring polynomials allows us to simplify expressions and solve for the roots. It's especially helpful when finding x-intercepts. Let's look at how each polynomial in the exercise is factored.
For the first function \( y = x^3 - x^2 \), we observe that there is a common factor of \( x^2 \). By factoring it out, we get \( y = x^2(x - 1) \).
For the second function \( y = (x-2)^3 - (x-2)^2 \), again we notice a common factor, \((x-2)^2 \). Factoring it out, we obtain \( y = (x-2)^2((x-2) - 1) \), which simplifies to \( y = (x-2)^2(x - 3) \).
Factoring polynomials often requires identifying the greatest common factor (GCF) and applying it to simplify the expression. Practice this skill to gain proficiency in polynomial problems.

Finding x-intercepts

Finding x-intercepts involves setting the function equal to zero and solving for x. The x-intercepts are the points where the graph crosses the x-axis.
For the function \( y = x^2(x - 1) \), setting \( y \) to zero gives us two factors: \( x^2 \) and \( x-1 \). Thus, setting \( x^2 = 0 \) gives us \( x = 0 \). Setting \( x-1 = 0 \) gives us \( x = 1 \). Therefore, the x-intercepts are \( x = 0 \) and \( x = 1 \).
For the function \( y = (x-2)^2(x-3) \), setting \( y \) to zero gives three factors, \( (x-2)^2 \) and \( x-3 \). Solving \( (x-2)^2 = 0 \) gives us \( x = 2 \), and solving \( x-3 = 0 \) gives us \( x = 3 \). Thus, the x-intercepts are \( x = 2 \) and \( x = 3 \).
The process of finding x-intercepts is valuable for analyzing the behavior of functions and understanding their graphical representation.