Problem 70 - Calculus Single Variable

Question

Find a function \(f\) whose graph is the given curve \(\mathcal{C}\). \(\mathcal{C}\) is obtained by translating the graph of \(y=x^{3}+2 x\) down 3 units and 2 units to the right.

Step-by-Step Solution

Answer

The function is \(f(x) = x^3 - 6x^2 + 14x - 15\).

Step 1: Understanding the Original Function

We start with the original function given as part of the curve: \(y = x^{3} + 2x\). This is the function before any translations, graphically representing a cubic function with a linear term.

Step 2: Horizontal Translation

The graph is translated 2 units to the right. To achieve a horizontal translation to the right, we replace \(x\) in the function with \(x - 2\). This results in the intermediate function \(y = (x - 2)^3 + 2(x - 2)\).

Step 3: Vertical Translation

After translating the graph to the right, we perform a downward vertical translation by reducing the whole function by 3 units. Thus, we modify the intermediate function to be: \(y = (x - 2)^3 + 2(x - 2) - 3\).

Step 4: Final Simplification of the Function

Now, let's expand and simplify the function from step 3. Start by expanding \((x - 2)^3\) and \(2(x - 2)\). \((x - 2)^3 = x^3 - 6x^2 + 12x - 8\) and \(2(x - 2) = 2x - 4\). Adding these together and simplifying yields \(x^3 - 6x^2 + 14x - 15\). Thus, the function \(f(x)\) that represents the translated graph is \(f(x) = x^3 - 6x^2 + 14x - 15\).

Key Concepts

Function Translation

In the world of mathematics, function translation plays a pivotal role in graph manipulation. When translating a function, we shift its graph on the coordinate plane without altering its shape or direction. This helps us understand how various changes to the equation impact the graphed curve. Translations are of two main types: horizontal and vertical.

• To translate horizontally, we alter the input value, which involves modifications inside the function's argument. This shifts the graph along the x-axis.
• To translate vertically, we adjust the output value, which means changing the value of the function as a whole. This results in a shift along the y-axis.

In our example, both types were applied to transform the initial cubic function into its final form.

Horizontal Translation

Horizontal translation involves shifting the graph left or right on the x-axis. We achieve this by altering the input variable within the function. When translating to the right, we subtract from the input value. Conversely, adding to the input value shifts the graph to the left.For our cube function example:

• The original function is given by: \( y = x^3 + 2x \).
• To translate the graph 2 units to the right, replace every \(x\) with \(x - 2\).
• This adjustment transforms the function to: \( y = (x - 2)^3 + 2(x - 2) \).

This change only affects how the graph moves along the x-axis, preserving the graph's general shape and orientation.

Vertical Translation

Vertical translation refers to moving the graph up or down along the y-axis. This process involves directly adding or subtracting a constant to the entire function. If we add a constant, the graph moves upwards; if we subtract, it moves downwards.Using the translated function from our earlier step:

• The function after the horizontal translation was \( y = (x - 2)^3 + 2(x - 2) \).
• To move the graph down by 3 units, we decrease the entire expression by 3.
• This operation modifies the function to : \( y = (x - 2)^3 + 2(x - 2) - 3 \).

The graph maintains its shape, but it moves lower on the coordinate plane by the specified amount.

Graphing Functions

Graphing functions involves plotting their equations on the coordinate plane, providing visual representations of the relationships defined by the equations. For a cubic function, the graph typically features a distinctive S-shaped curve.Key aspects when graphing transformed functions include:

• Understanding the original function's shape, which in this case is a cubic curve with a linear term.
• Applying horizontal and vertical translations as needed, which shifts the graph without changing its overall structure.
• Simplifying the equation after translations can help in verifying its new form for accurate graphing.

For the given example, after simplifying the translated function: \( f(x) = x^3 - 6x^2 + 14x - 15 \), we can plot these modifications to see exactly how the original graph has shifted across the plane.