Question

Use the graph of \(y=x^{3}\) to sketch the graph of the function. $$f(x)=(x+3)^{3}$$

Short answer

The graph of the given function \(f(x)=(x+3)^{3}\) is a cubic function that has been shifted three units to the left from the standard cubic function \(y = x^{3}\). The shape of the curve of this graph will remain the same as that of a regular cubic function.

Step-by-step solution

Identify the transformation

In the function \(f(x)=(x+3)^{3}\), the \(+3\) inside the brackets with the \(x\) indicates a shift or translation of the graph. This shift is horizontal and is opposite to the direction of the sign, meaning a shift of three units to the left, often noted as a horizontal shift of \(-3\). This tells us that every point on the graph of \(y = x^{3}\) is moved three units to the left to form the graph of \(f(x)\).

Plot the new function

We can then draw the graph of \(f(x)=(x+3)^{3}\). Start with the basic graph of the cubic function \(y = x^{3}\). Now shift every point on this graph three units to the left. For instance, points that were originally at (1,1), (2, 8), (0, 0) would be shifted to (-2, 1), (-1, 8), and (-3, 0) respectively. The same should be done for the points on the negative side of the x-axis. The shape of the curve remains the same - only the position changes due to the shift.

Check your work

After sketching this graph, it's a good idea to double-check the work: The points should have been moved three units to the left from the original cubic function.

Key concepts

Cubic Functions

Cubic functions are interesting because of their unique cubic term, expressed as \( y = x^3 \). These functions create classic S-shaped curves that can stretch infinitely in both the positive and negative directions. As you increase the power of the function, the graph tends to grow steeper more rapidly. This behavior gives cubic functions their distinctive look.

A simplified form \( y = x^3 \) has critical points at the origin (0,0), causing the curve to be symmetrical around this point. In more complex cubic functions, additional linear terms can cause shifts and stretches. Despite these manipulations, the fundamental cubic shape remains consistent, which allows us to identify it easily.

When understanding how transformations like shifts or stretches affect cubic functions, it's helpful to start with a sketch of the basic form. Recognize the starting point for any other changes that might be requested as part of the function modifications.

Horizontal Shifts

Horizontal shifts are transformations that move the graph of a function left or right. These occur when a constant is added or subtracted inside the function’s argument—particularly when dealing with polynomials. - To move left, you add a positive number inside the parenthesis. For example, in \( f(x) = (x + 3)^3 \), the graph shifts 3 units to the left. - To move right, you subtract a positive number. So, if it were \( f(x) = (x - 3)^3 \), the graph would shift 3 units to the right.
This technique let's you take the basic graph of \( y = x^3 \) and adjust it to form \( f(x) = (x + 3)^3 \) by translating every point three units to the left. Understanding these shifts is crucial for graphing functions quickly and accurately. A good strategy is practicing by shifting different types of functions and checking if the final graph matches the expected outcome.

Function Graphing

Graphing functions effectively blends intuition and technical skill. It's a way to visualize mathematical equations in a more tangible format. Start with the base graph of the original function (like \( y = x^3 \) for cubics) and then apply transformations such as shifts.

Some tips to achieve precise graphing:

• Begin by calculating key points. For a cubic function, this includes intercepts and a few points along its path.
• Apply transformations such as shifts, stretches, or reflections sequentially. Check the impact of each step before moving to the next.
• Draw a sketch of the transformed function cautiously. Double-check points to ensure accuracy, particularly after a horizontal shift.

Comfort and accuracy in graphing come with practice. It's like learning a new language—initially challenging, but over time, becomes second nature. Regularly visualize how the algebra translates into the graphical world, and you'll make this process smoother.