Problem 19 - 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. \(g(x)=\sqrt{x-3}+1\)

Step-by-Step Solution

Answer

The function \(g(x)=\sqrt{x-3}+1\) is a transformation of the basic function \(f(x)=\sqrt{x}\), involving a shift three units to the right and one unit upwards. The graph of \(g\) starts from the point (3,1) and increases like \(f\) as \(x\) increases.

Step 1: Identify Basic Function and Transformation

The basic function here is \(f(x)=\sqrt{x}\), a square root function. Our transformed function \(g(x)=\sqrt{x-3}+1\) involves shifting. The \(-3\) inside the square root function suggests a horizontal shift to the right by three units. The \(+1\) outside the square root suggests a vertical shift up by one unit.

Step 2: Sketch the Basic Function

Plot the square root function, \(f(x)=\sqrt{x}\). This function starts from the origin (0,0) and increases as \(x\) increases, creating a curve in the first quadrant.

Step 3: Apply Transformations and Sketch \(g\)

Start by shifting the basic square root curve three units to the right. This accounts for the \(\sqrt{x-3}\). Then, shift the result one unit upwards to account for the \(+1\). The curve of \(g(x) = \sqrt{x-3}+1\) starts from the point (3,1) and continues similarly to the basic square root function.

Step 4: Verify with a Graphing Utility

Use a graphing calculator or online tool to verify the accuracy of the sketch. Input the function \(g(x)=\sqrt{x-3}+1\) and check that the graph matches the hand-drawn sketch.

Key Concepts

Square Root Function

The square root function, mathematically represented as \(f(x) = \sqrt{x}\), is a fundamental mathematical concept. This function produces values that are consistent with the non-negative square roots of corresponding \(x\) values. In simpler terms, for every input, \(x\), we find the number that, when squared, gives us \(x\).

• **Behavior**: The graph of \(f(x) = \sqrt{x}\) starts at the origin (0,0).
• **Direction**: It increases steadily but slowly, creating a curve in the first quadrant.
• **Domain and Range**: The domain, or possible input values, of this function is non-negative \(x\)-values, \([0, \infty)\). The range, or possible output values, is also non-negative \(y\)-values, \([0, \infty)\).

The square root function is essential for understanding transformations, as its simple shape makes it easier to visualize shifts and changes.

Horizontal Shift

A horizontal shift involves moving a function's graph left or right. In the transformation from \(f(x) = \sqrt{x}\) to \(g(x) = \sqrt{x-3}\), we implement a horizontal shift. This specific shift is guided by the \(x - 3\) inside the square root:

• **Direction of Shift**: Since we're subtracting 3 from \(x\), the graph shifts to the right by 3 units.
• **Effect on Graph**: This transformation adjusts all the points on the graph of \(f(x) = \sqrt{x}\), shifting them horizontally. The starting point initially at (0,0) moves to (3,0).

This concept plays a crucial role in modifying graphs, allowing them to be repositioned along the \(x\)-axis without changing their basic shape.

Vertical Shift

The vertical shift moves a graph up or down, altering its position along the \(y\)-axis. When transitioning from \(f(x) = \sqrt{x-3}\) to \(g(x) = \sqrt{x-3} + 1\), a vertical shift is applied. Here the transformation is driven by the addition of 1 outside the square root:

• **Direction of Shift**: Since we are adding 1, the whole graph shifts upwards by one unit.
• **Effect on Graph**: The graph, now horizontally shifted to the right, is then moved up, turning the starting point from (3,0) to (3,1).

Vertical shifts are straightforward adjustments that allow visual enhancements to the function's graph without affecting its \(x\)-coordinate, modifying only the height of the function.