Use the​ x-cubed tool to graph the​ x-cubed function

Use the​ x-cubed tool to graph the​ x-cubed function. Update the function to have a Vertical Stretch of​ 3, a Horizontal Stretch of​ 1, a Vertical Shift of​ 0, a Horizontal Shift of​ 5, and a Reflection over the​x-axis.

Use the​ x-cubed tool button

on the graphing palette to place the function on the graph. Then update the Vertical​ Stretch, Horizontal​ Stretch, Vertical​ Shift, Horizontal​ Shift, and Reflect over​ x-axis interactions to have the values noted above.

The Correct Answer and Explanation is :

To graph the transformed cubic function with the given transformations, follow these steps:

Step 1: Understand the Base Function

The parent function is:
[
f(x) = x^3
]

Step 2: Apply Transformations

We need to apply the following transformations:

1. Vertical Stretch by 3: Multiply the function by 3:
   [
   f(x) = 3x^3
   ]
2. Horizontal Stretch by 1: A horizontal stretch by a factor of ( a ) is represented as ( f(x) = (1/a \cdot x)^3 ). Since the factor is 1, the function remains unchanged.
3. Horizontal Shift by 5: This means shifting the graph to the right by 5 units. Replace ( x ) with ( (x – 5) ):
   [
   f(x) = 3(x – 5)^3
   ]
4. Vertical Shift of 0: Since there’s no vertical shift, the function remains the same.
5. Reflection over the x-axis: To reflect over the x-axis, multiply the entire function by ( -1 ):
   [
   f(x) = -3(x – 5)^3
   ]

Step 3: Final Transformed Function

[
f(x) = -3(x – 5)^3
]

Explanation of Transformations

• The vertical stretch by 3 makes the graph steeper, meaning it grows faster in magnitude.
• The horizontal stretch of 1 keeps the function’s width unchanged.
• The horizontal shift of 5 moves the function 5 units to the right.
• The vertical shift of 0 means there is no movement up or down.
• The reflection over the x-axis flips the function upside down, making all positive y-values negative and vice versa.

Graphing Instructions

1. Plot the parent function ( x^3 ).
2. Shift it 5 units right.
3. Apply a vertical stretch by multiplying y-values by 3.
4. Reflect the graph over the x-axis.
5. The final function should pass through (5,0) and follow the transformed cubic shape.

Now, let’s generate a graph of this function.

Here is the graph of the function ( f(x) = -3(x – 5)^3 ). The key transformations are applied:

• Shifted 5 units to the right, moving the inflection point from (0,0) to (5,0).
• Vertically stretched by 3, making the curve steeper.
• Reflected over the x-axis, flipping the original cubic shape upside down.

This visualization confirms how the transformations affect the parent function. Let me know if you need further clarification! 🚀