Problem 24 - Algebra and Trigonometry

Question

23–26 ? Explain how the graph of g is obtained from the graph of f. (a) \(f(x)=x^{3}, \quad g(x)=(x-4)^{3}\) (b) \(f(x)=x^{3}, \quad g(x)=x^{3}-4\)

Step-by-Step Solution

Answer

(a) Shift right 4 units; (b) Shift down 4 units.

Step 1: Examine the Function g(x) in (a)

Given that \(g(x) = (x-4)^3\), we compare it to \(f(x) = x^3\). This equation represents a horizontal shift. The function \(g(x)\) is obtained by shifting \(f(x)\) 4 units to the right. This shift is due to the expression \((x-4)\) inside the cube, which indicates a movement in the x-direction.

Step 2: Examine the Function g(x) in (b)

For \(g(x) = x^3 - 4\), compare it to \(f(x) = x^3\). This equation represents a vertical shift. The graph of \(g(x)\) is obtained by shifting the graph of \(f(x)\) downward by 4 units. The subtraction of 4 outside the cubic term indicates a downward shift in the y-direction.

Key Concepts

Horizontal Shift

In mathematical terms, a horizontal shift occurs when the graph of a function moves left or right along the x-axis. This type of transformation is crucial as it can change the location of the graph without altering its shape. A common way to identify a horizontal shift in the function transformation process is by looking inside the function.

• If you see a term like "+c" or "-c" inside the function, it signifies a horizontal shift. For example, in the function \(g(x) = (x-4)^3\), the expression "\(x-4\)" suggests a horizontal shift.
• The key here is to recognize that "x-4" indicates a shift of 4 units to the right. The negative sign means you shift right, opposite to what it might suggest from the subtraction.
• Always remember, inside the function affects the horizontal direction, and opposites attract! "-4" causes a move to the right by 4 units.

Processing these changes can significantly adjust how one models and understands the behavior of graphs in various functions.

Vertical Shift

Vertical shifts involve moving the graph of a function up or down along the y-axis. This is a straightforward concept that changes the graph's position without any impact on its shape.

• You can identify a vertical shift when you notice a number added or subtracted outside the main function. For instance, with \(g(x) = x^3 - 4\), the "-4" indicates a vertical shift.
• Subtracting 4 means shifting the graph downward by 4 units, while adding a number would mean moving upwards.
• Vertical shifts are simplest to visualize because they directly change the y-coordinates of every point of the graph, making it easy to predict the outcome.

Vertical shifts simply move the graph up or down, allowing for quick recalibrations in the function's graph.

Graph Translation

Graph translation refers to the overall movement of a graph in the coordinate plane. It combines both horizontal and vertical shifts, emphasizing the transformation of the entire graph.

• A translation doesn't warp or squeeze the graph; it merely relocates it to a different region of the plane.
• This action can involve moving the graph in any direction, dictated by horizontal and vertical shifts.
• In our cases: \(g(x) = (x-4)^3\) translates \(f(x)\) 4 units to the right, whereas \(g(x) = x^3 - 4\) moves it downward by 4 units.

Graph translation thus allows us to alter the position and orientation of graphs effectively, enabling refined adjustments to better fit data or other functions.