Problem 61 - Algebra 2

Question

Critical Thinking Recall that each family of functions has a simplest function called the parent function. a. Compare the graphs of \(y=x^{3}\) and \(y=x^{3}+4\) Describe how the graph of \(y=x^{3}+4\) relates to the graph of \(y=x^{3}\) . b. Compare the graphs of \(y=x^{3}\) and \(y=4 x^{3} .\) Describe how the graph of \(y=4 x^{3}\) relates to the graph of \(y=x^{3}\) . c. Identify the parent function among the functions in parts (a) and (b).

Step-by-Step Solution

Answer

The graph of \(y = x^3 + 4\) is a vertical shift 4 units up of the graph of the parent function \(y = x^3\). The graph of \(y = 4x^3\) is a vertical stretch by a factor of 4 of the parent function. The parent function is \(y = x^3\).

Step 1: Understanding the Vertical Shift

The graph of the function given by \(y = x^3 + 4\) is the result of shifting the graph of the parent function \(y = x^3\) upwards by 4 units. This kind of transformation is called a vertical shift.

Step 2: Analysing the Vertical Stretch

Comparing the graph of \(y = 4x^3\) with the parent function \(y = x^3\), we can see that the graph of \(y = 4x^3\) is a vertical stretch by a factor of 4. This means that for any value of x, the value of y in the function \(y = 4x^3\) is four times larger than that in the parent function.

Step 3: Identifying the Parent Function

The parent function among the given functions is \(y = x^3\). This is because the other two functions are transformations of this parent function, with the former being a vertical shift and the latter being a vertical stretch.

Key Concepts

Graph Transformations

Graph transformations refer to various operations that modify the appearance of a graph without changing its fundamental shape or properties. These operations include shifting, stretching, compressing, and reflecting the graph of a function.

Imagine you're looking at a painting on a wall; graph transformations are akin to moving that painting up, down, stretching it taller, or squishing it wider, but the painting itself remains the same. The crucial aspect to understand is that each transformation has specific rules.

• A vertical shift moves the graph up or down along the y-axis.
• A horizontal shift moves the graph left or right along the x-axis.
• Stretching or compressing a graph involves pulling or squeezing it vertically or horizontally.
• Reflection involves flipping the graph across a line, like the x-axis or y-axis.

These transformations allow us to easily generate new functions from a basic parent function, which is a simpler form of the function that retains its most critical features.

Vertical Shift

A vertical shift is one of the simplest graph transformations, moving a graph up or down in a uniform manner. It doesn't alter the shape of the graph but changes its position on the coordinate plane.

When we add or subtract a number from the function's formula, we are applying a vertical shift.

Addition results in an upward shift

For instance, if you have the parent function given by \(y = x^3\), and you add 4 to get \(y = x^3 + 4\), the entire graph shifts up by 4 units. It's like giving every point on the graph a little boost upwards.

Subtraction results in a downward shift

Conversely, if we subtract a number, say 5, to form \(y = x^3 - 5\), the graph shifts down by 5 units.

Vertical shifts are commonly used to adjust the starting point of a graph, which can be crucial in real-world contexts where the baseline isn’t zero.

Vertical Stretch

A vertical stretch or compression changes the 'tallness' or 'flattening' of a graph without affecting its x-values. When a graph is vertically stretched, it becomes taller, and each y-value is multiplied by a factor greater than 1.

Consider the function \(y = x^3\) undergoing a vertical stretch by a factor of 4, yielding the function \(y = 4x^3\). In this case,

Every y-value becomes 4 times larger

This means for any input x, the corresponding y-value is quadrupled, hence the graph appears 'taller' compared to the parent function. This operation is particularly useful in modeling situations where quantities increase at a faster rate.

Vertical Compression

In contrast, if the factor is between 0 and 1, like in \(y = \frac{1}{2}x^3\), the graph is compressed and appears 'flatter'.

Understanding these transformations helps not just in algebra but in calculus as well, where they play a critical role in analyzing function behavior and in practical applications like signal processing or designing structures.