Problem 49 - A Graphical Approach to College Algebra

Question

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$f(x)=(x-1)^{3}$$

Step-by-Step Solution

Answer

Shift the graph of \(y=x^3\) right by 1 unit to sketch \(y=(x-1)^3\).

Step 1: Identify the Parent Function

Begin by recognizing the parent function of the given function. The function \(f(x) = (x - 1)^3\) is a transformation form of the parent function \(g(x) = x^3\), which is a cubic function.

Step 2: Determine the Transformation

The function \(f(x) = (x - 1)^3\) indicates a horizontal shift of the parent function. Specifically, the \((x - 1)\) term suggests that there is a shift of 1 unit to the right along the x-axis. This means the graph of \(x^3\) is translated rightward by 1 unit.

Step 3: Sketch the Parent Function

Start by sketching the graph of the parent function \(g(x) = x^3\). This is the basic cubic curve which passes through the origin \((0,0)\), with similar points such as \((-1, -1)\), \((1, 1)\), \((-2, -8)\), and \((2, 8)\), reflecting its characteristic shape of an "S" curve.

Step 4: Apply the Transformation

Take the graph of \(g(x) = x^3\) and shift each point 1 unit to the right. The origin point \((0, 0)\) moves to \((1, 0)\). Likewise, point \((1, 1)\) moves to \((2, 1)\), point \((-1, -1)\) moves to \((0, -1)\), and so forth. Re-sketch the curve based on these newly positioned points.

Step 5: Finalize the Graph

Ensure your graph accurately reflects the cubic shape, but translated 1 unit to the right. The inflection point at \((1, 0)\) is a key characteristic.

Key Concepts

Cubic Functions

Cubic functions are a type of polynomial where the degree is three, meaning the highest power of the variable is cubed. The general form of a cubic function is \(f(x) = ax^3 + bx^2 + cx + d\), where coefficients \(a\), \(b\), \(c\), and \(d\) determine the specific shape and position of the curve. Cubic functions produce graphs that have a distinctive "S" shaped curve, commonly called a cubic curve. These graphs have an inflection point, where the curve changes from concave upwards to concave downwards, or vice versa.
Cubic functions are known for always crossing the x-axis three times, two times, or even just once, depending on whether the roots are real or complex. In the exercise provided earlier, the parent function of \(f(x) = (x - 1)^3\) is \(g(x) = x^3\), a classic cubic equation without any quadratic or linear terms, making the analysis of its transformation straightforward and ideal for fundamental learning.

Parent Functions

Parent functions are the simplest form of functions from which other functions in the same family can be derived through transformations. For each type of function, there is one basic parent function.

• Linear functions have the parent function \(f(x) = x\).
• Quadratic functions have the parent function \(f(x) = x^2\).
• Cubic functions have the parent function \(f(x) = x^3\), as it is for our current problem.

The concept of parent functions is fundamental when dealing with graph transformations. Recognizing a parent function helps to understand how transformations will alter the graph of the function. For example, in the case of cubic functions, transforming \(f(x) = x^3\) gives insights into more complex functions like \(f(x) = (x - 1)^3\). Understanding how the basic "S" shape shifts and stretches allows students to predict graph changes without plotting numerous points.

Horizontal Shifts

Horizontal shifts are a type of transformation that moves a graph left or right along the x-axis without changing its shape. This transformation is represented in the function equation of the form \(f(x) = (x - h)^3\), where \(h\) is the number of units the graph is shifted.
Horizontal shifts can be identified as such:

• If \(h > 0\), the entire graph shifts to the right by \(h\) units.
• If \(h < 0\), the graph shifts to the left by \(-h\) units.

In our example, \(f(x) = (x - 1)^3\) indicates a shift 1 unit to the right. This shift moves each point of the parent graph \(g(x) = x^3\) rightward, changing the equation without altering its core characteristics. The process is intuitive—replace the \(x\)-coordinates of key graph points by adding \(h\), creating a new set of points that describe the shifted graph. Understanding horizontal shifts is key in mastering graph transformations, allowing visualization of changes without recalculating values extensively.