Problem 24 - Calculus for Biology and Medicine

Question

Explain how the following functions can be obtained from \(y=x^{3}\) by basic transformations: (a) \(y=x^{3}-1\) (b) \(y=-x^{3}-1\) (c) \(y=-3(x-1)^{3}\)

Step-by-Step Solution

Answer

(a) Vertical shift down by 1; (b) Reflect over x-axis, shift down by 1; (c) Translate right by 1, stretch vertically by 3, reflect over x-axis.

Step 1: Understanding Transformations

The parent function given is \(y = x^3\). We need to apply transformations to achieve each specified function. Transformations include translations (up, down, left, right), reflections (across axes), dilations (stretches or compressions), and combinations of these.

Step 2: Analyzing Transformation for (a)

For the function \(y = x^3 - 1\), identify the transformation from \(y = x^3\). This transformation is a vertical translation downward by 1 unit, which subtracts 1 from the whole function.

Step 3: Analyzing Transformation for (b)

For \(y = -x^3 - 1\), notice the transformation involves two operations: reflecting \(y = x^3\) across the x-axis (indicated by the negative sign in front of \(x^3\)) and then translating the result downward by 1 unit, which again involves subtracting 1.

Step 4: Analyzing Transformation for (c)

In \(y = -3(x-1)^3\), multiple transformations are present. First, there is a horizontal translation to the right by 1 unit shifting the cube root into \((x - 1)^3\). The function is then vertically stretched by a factor of 3 (multiplying the cube by 3), and finally, reflected across the x-axis due to the negative sign, changing the sign of all terms.

Key Concepts

Vertical Translation

Vertical translation is a straightforward but significant transformation. It involves shifting the entire graph of a function up or down along the y-axis without changing its shape. Here's how it works:

• Adding a constant to a function, like in \(y = x^3 + c\), moves the graph up by \(c\) units.
• Subtracting a constant, as seen in \(y = x^3 - 1\), shifts the graph down by 1 unit.

For example, if you have the parent function \(y = x^3\), transforming it to \(y = x^3 - 1\) requires a vertical translation downward by one unit on the y-axis. This adjustment affects the y-values of all points on the graph but leaves the x-values unchanged.

Horizontal Translation

Horizontal translation shifts the graph of a function left or right along the x-axis. Unlike vertical translations, which involve adding or subtracting constants, horizontal translations are achieved through changes inside the function's variable:\(x\).

• When you have \(y = (x - 1)^3\), like in the function \(y = -3(x-1)^3\), it's moved right by 1 unit.
• A term \(x + c\) translates the function left by \(c\) units.

It's crucial to note these changes shift the location of the graph's features along the x-axis, making it important to adjust the interpretation of the function accurately.

Reflecting Functions

Reflecting a function typically involves flipping it across one of the axes, modifying how it appears visually. For reflection across the x-axis, consider what happens when you multiply the entire function by \(-1\). Basics of reflections include:

• If \(y = f(x)\), then \(y = -f(x)\) reflects across the x-axis.
• Example: Starting with \(y = x^3\), reflecting it gives \(y = -x^3\).

Combining reflection with other transformations, like in \(y = -3(x-1)^3\), involves careful handling of signs and transformations to ensure the graph's correct orientation.

Parent Function

The parent function serves as your starting point before applying transformations. It's the simplest form, providing a basic reference template for function graphs. For polynomials like cubes, the parent function is \(y = x^3\). Essential qualities of a parent function include:

• Simplicity: The basic form, with no transformations.
• Comparison: Offers a baseline for understanding transformations.

By understanding the parent function, it's easier to visualize and calculate the effects of each transformation, as transformations derive from this baseline form.