Problem 17 - College Algebra and Calculus: An Applied Approach

Question

Describe the sequence of transformations from \(f(x)=\sqrt{x}\) to \(g\). Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=\sqrt{x-3}\)

Step-by-Step Solution

Answer

The function \(g(x)\) represents a transformation of the square root function \(f(x)\) where the graph is shifted 3 units to the right. This is denoted by the \(\sqrt{x-3}\) from f(x). The graphing of g(x) will look like the graph of f(x) but starting not at the origin (0,0) but at the point (3,0).

Step 1: Understand the basic root function

The base function here is \(f(x)\), which is the basic square root function. Its graph is a half parabola rotated by 90 degrees counter-clockwise. It starts at the origin \((0,0)\) and grows towards positive x and y axes.

Step 2: Understand the transformation

The function \(g(x)\) is obtained from \(f(x)\) by replacing \(x\) with \((x-3)\). This corresponds to a shift to the right. In general, if you have a function \(f(x)\) and you replace \(x\) with \((x-a)\), where \(a\) is a positive number, this will shift the graph \(a\) units to the right.

Step 3: Sketch the graph of \(g(x)\)

To sketch the graph of \(g(x)\), start by sketching the basic square root function like mentioned in Step 1. Then, shift the graph 3 units to the right as per Step 2. Make sure to verify this process with a graphing tool.

Key Concepts

Square Root Function

The square root function is one of the most fundamental functions in mathematics. It is typically denoted as \( f(x) = \sqrt{x} \), where \( x \) is a non-negative number. The graph of the square root function can be visualized as a curve starting from the origin, (0,0), and incrementally moving upwards to the right in a shape resembling half of a parabola rotated counter-clockwise.
This type of curve only covers the first quadrant of the Cartesian plane since the square root of a negative number is not a real number in the standard real number system.

• The function is defined only for \( x \geq 0 \).
• It is ever-increasing, meaning as \( x \) increases, \( \sqrt{x} \) also increases.
• The rate of increase in the function value diminishes as \( x \) gets larger.

This foundational understanding enables us to transform this basic graph into more complex forms by applying various transformations, as shown in the next section.

Horizontal Shift

Horizontal shifts are fundamental transformations involving the movement of a graph along the x-axis. They occur when you take a function and modify its input value directly. In our exercise, we begin with the function \( f(x) = \sqrt{x} \). When a transformation of the form \( g(x) = \sqrt{x - a} \) is applied, it results in shifting the graph of the original function \( a \) units to the right.
This is because each x-value on the f(x) graph is replaced by another value that is \( a \) units larger, effectively delaying the graph's start by \( a \) units.
Here’s how you can visualize it:

• If \( a = 3 \), then we obtain \( g(x) = \sqrt{x - 3} \).
• The graph shows the identical shape to \( f(x) \) but it now begins at \( x = 3 \).
• This entire curve moves seamlessly rightward, maintaining its integrity and position relative to the y-axis.

This transformation is essential in graph manipulation, enabling diverse functional behaviors within mathematical modeling.

Graph Plotting

Graph plotting is a crucial skill for visualizing and understanding mathematical functions. With the given transformations, plotting \( g(x) = \sqrt{x-3} \) becomes straightforward once the transformations are understood. Begin by plotting the graph of the basic square root function \( f(x) = \sqrt{x} \).
Now apply the horizontal shift:

• Shift the entire graph rightward by 3 units.
• Ensure the new starting point of the graph is \( (3,0) \) instead of \( (0,0) \).
• Keep the shape unchanged; it still opens to the right and lies in the first quadrant.

This new graph visually confirms our earlier algebraic transformation. It's always helpful to use graphing utilities to double-check your manually drawn graphs. These tools provide precision and further insights into how the transformations work. They verify accuracy and enhance comprehension by allowing students to see immediate feedback and potential errors in their initial sketches.