Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. What is the function that is graphed after the graph of \(y=\sqrt[3]{x}\) is shifted left 4 units and up 9 units?

Short answer

y = \sqrt[3]{x + 4} + 9

Step-by-step solution

- Understand the base function

The base function given is \( y = \sqrt[3]{x} \) which is the cube root of x.

- Leftward shift

To shift the graph left by 4 units, you replace every \( x \) in the function with \( x + 4 \). So, \( y = \sqrt[3]{x} \) becomes \( y = \sqrt[3]{x + 4} \).

- Upward shift

To shift the graph up by 9 units, you add 9 to the function. Hence, \( y = \sqrt[3]{x + 4} \) becomes \( y = \sqrt[3]{x + 4} + 9 \).

Key concepts

cube root functions

Cube root functions involve the cube root of a variable, most commonly represented as \( y = \sqrt[3]{x} \). These functions are unique because they can handle both positive and negative values of \ x \ due to the properties of cube roots. For instance, \( \sqrt[3]{8} = 2 \) and \( \sqrt[3]{-8} = -2 \).

Understanding cube root functions is crucial because they stretch and compress the graph in unique ways, which is different from square root functions that only work for non-negative x-values. The shape of the cube root function graph resembles an S-curve, smoothly passing through the origin (0,0).

With the base function \( y = \sqrt[3]{x} \), the input and output values can be explicitly evaluated to understand the behavior of the graph better.

shifting graphs

Shifting graphs involves moving the entire graph of a given function either horizontally, vertically, or both. This transformation does not alter the shape of the graph but changes its position in the coordinate plane.

There are two main types of shifts: horizontal and vertical.

These shifts are applied to the function equation directly and help in analyzing and comparing different graphical representations of similar functions. Practice with these shifts helps in visualizing function transformations more easily.

horizontal shift

Horizontal shifts move the graph of a function left or right along the x-axis. They are achieved by adding or subtracting a constant from the x-component of the function.

For example, in the case of our base function \( y = \sqrt[3]{x} \), to shift the graph left by 4 units, we replace every \ x \ with \( x + 4 \). This results in the transformed function \( y = \sqrt[3]{x + 4} \).

If you want to shift the graph to the right, you subtract a constant instead. For instance, shifting the graph right by 3 units results in \( y = \sqrt[3]{x - 3} \).

Understanding this concept helps in manually graphing functions and quickly identifying the movement from the base position.

vertical shift

Vertical shifts move the graph of a function up or down along the y-axis. They are achieved by adding or subtracting a constant from the entire function.

For example, let’s use our horizontally modified function \( y = \sqrt[3]{x + 4} \). To shift this graph up by 9 units, you simply add 9 to the entire function, resulting in \( y = \sqrt[3]{x + 4} + 9 \).

Conversely, to shift the graph down, you would subtract a constant. Shifting the same function down by 5 units would change it to \ y = \sqrt[3]{x + 4} - 5 \.

Vertical shifts help in adjusting the graph's height on the coordinate plane, which can be particularly useful for function comparison or aligning transformations with specific contexts.