Problem 15 - Elementary and Intermediate Algebra

Question

a. The graph of \(f(x)=(x+4)^{3}\) is the same as the graph of \(f(x)=x^{3}\) except that it is shifted _____ units to the _____. b. The graph of \(f(x)=x^{3}+4\) is the same as the graph of \(f(x)=x^{3}\) except that it is shifted _____ units _____.

Step-by-Step Solution

Answer

a. 4 units to the left; b. 4 units upward.

Step 1: Understanding Horizontal Shifts

In the function \(f(x) = (x+4)^3\), we see \((x + 4)\) inside the cube operation. This indicates a horizontal shift. Remember that \( f(x + c) \) shifts the graph left by \(c\) units. Therefore, \( (x+4)^3 \) shifts the graph 4 units to the left.

Step 2: Analyzing Vertical Shifts

For the function \(f(x) = x^3 + 4\), the \(+ 4\) is outside of the original function \(x^3\). This indicates a vertical shift. The addition of \(+4\) to the function causes it to shift upward by 4 units.

Key Concepts

Horizontal Shifts

Horizontal shifts involve moving a graph left or right on the coordinate plane. This shift occurs when a constant is added or subtracted inside the function's input. For instance, observe the function \(f(x) = (x+4)^3\). When we see \((x + 4)\) inside the cube operation, it tells us there's a horizontal shift involved.

A critical point to remember is that the expression \(f(x + c)\) shifts the graph of \(f(x)\) to the left by \(c\) units. The reason for this seemingly counter-intuitive direction is due to the negative sign in front of \(c\).
* So, substituting \(+4\) makes us shift 4 units to the left, even though it feels like it should go the other way.
Understanding horizontal shifts helps to quickly alter your graph without redrawing it entirely.

Vertical Shifts

Vertical shifts are simpler to understand than horizontal shifts. These shifts occur when a constant is added or subtracted outside of a function, directly affecting its output. Take, for example, the function \(f(x) = x^3 + 4\). Here, the \(+4\) is added outside of the base function \(f(x) = x^3\).

Intuitively, adding a positive number causes the entire graph to move up. Conversely, subtracting a number would move the graph down. Let’s break it down further:

• In \(f(x) = x^3 + 4\), the graph will move 4 units upward since 4 is added.
• A similar function like \(g(x) = x^3 - 2\) would shift the graph downward by 2 units, due to the subtraction of 2.

Vertical shifts are straightforward and offer an efficient way to change your graph's position along the y-axis.

Graph Transformations

Graph transformations cover a range of techniques used to modify the position and shape of a graph. Not only do they include horizontal and vertical shifts, but also reflections, stretches, and compressions. Understanding these transformations allows you to adjust and predict graph behavior with ease.

Focusing on shifts specifically:

• Horizontal shifts move a graph side-to-side with expressions like \(f(x + c)\) or \(f(x - c)\), shifting left or right accordingly.
• Vertical shifts raise or lower a graph along the y-axis, using terms such as \(f(x) + d\) or \(f(x) - d\).

Graph transformations are powerful because they allow for graph adjustment without complicated recalculations. With these tools, shifting a graph to better fit your data, for example, becomes a quick and intuitive process.