Problem 60 - College Algebra with Modeling and Visualization

Question

Use transformations to sketch a graph of \(f\). \(f(x)=2 \sqrt{(x-2)}-1\)

Step-by-Step Solution

Answer

Shift right 2, stretch vertically by 2, move down 1 unit.

Step 1: Identify the Base Function

The base function of the given equation is the square root function, which is \( g(x) = \sqrt{x} \). This is the starting point for graph transformations.

Step 2: Horizontal Shift

The function \( f(x) = 2 \sqrt{x-2} - 1 \) indicates a horizontal shift. Specifically, the \(x\) inside the square root is replaced by \(x-2\), which results in a shift to the right by 2 units compared to the base function \( g(x) = \sqrt{x} \).

Step 3: Vertical Stretch

The factor 2 in front of the square root \( 2\sqrt{x-2} \) scales the graph vertically by a factor of 2. This means that each \(y\)-value of the base function is multiplied by 2 when shifted.

Step 4: Vertical Shift

The term \(-1\) at the end of the function \( 2\sqrt{x-2} - 1 \) indicates a vertical shift downwards by 1 unit. This affects the entire graph from the previous step.

Step 5: Sketch the Graph

Start with the base graph of \( y = \sqrt{x} \). Shift it right by 2 units, then stretch it vertically by doubling the height of each point. Finally, shift the resulting graph downward by 1 unit. This is the sketch of \( f(x) = 2\sqrt{x-2} - 1 \).

Key Concepts

Square Root Function

The square root function forms the basis for our graph transformations. This function is defined as \( g(x) = \sqrt{x} \), a simple yet important mathematical function often seen in various applications. Its graph is unique and easily recognizable: it begins at the origin \( (0,0) \) and extends rightward, gradually increasing and forming a gentle curve. Unlike linear functions, the square root function only exists for non-negative values of \( x \), as negative values of \( x \) do not produce real numbers. Understanding the behavior of the base square root function is crucial when performing graph transformations, as all transformations revolve around altering this primary function.

Horizontal Shift

Horizontal shifts are transformations that move the graph of a function left or right in the plane. For the function \( f(x) = 2\sqrt{x-2} - 1 \), we observe this horizontal shift inside the square root, where \( x \) is replaced by \( x-2 \). This signals a rightward shift by 2 units.

• This occurs because the subtraction inside the function translates the graph in the opposite direction of the sign.
• As a result, every point on the original graph of \( g(x) = \sqrt{x} \) moves 2 units to the right.

The horizontal shift doesn’t affect the shape of the graph, only its position along the x-axis, making it an essential step in understanding where the transformation takes place.

Vertical Stretch

Vertical stretching affects the vertical dimensions of the graph of a function. With a function like \( f(x) = 2\sqrt{x-2} - 1 \), the coefficient 2 in front of the square root causes a vertical stretch.

• This means every point's height is doubled compared to the original height on the base curve \( g(x) = \sqrt{x} \).
• It transforms the function's rate of increase, making it rise more steeply as \( x \) increases.

Vertical stretches change the overall steepness but do not shift the graph left, right, up, or down. Understand that larger coefficients stretch the graph more, leading to a sharp incline.

Vertical Shift

Vertical shifts adjust the position of the entire graph up or down along the y-axis. In this instance, the function \( f(x) = 2\sqrt{x-2} - 1 \) incorporates a vertical shift due to the -1 at the end.

• This indicates a downward movement of the entire graph by 1 unit.
• Every point on the graph of \( 2\sqrt{x-2} \) after being stretched will be shifted down by 1 unit.

Such shifts do not alter the shape or angle of the graph, only its vertical location. Understanding vertical shifts helps you position your final graph correctly relative to the unshifted version.