Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand $$f(x)=2 \sqrt{x-2}-1$$

Step-by-Step Solution

Answer

Shift right 2 units, stretch vertically by 2, and shift down 1 unit.

Step 1: Identify the Parent Function

The parent function of the given function is the square root function, which is \( y=\sqrt{x} \). This function has a graph that starts at the origin \((0,0)\) and increases slowly, curving upwards to the right.

Step 2: Apply Horizontal Transformation

The expression inside the square root, \(x-2\), represents a horizontal shift. Specifically, this equation shifts the graph 2 units to the right. Therefore, the starting point \((0,0)\) of the parent function moves to \((2,0)\).

Step 3: Apply Vertical Stretch and Transformation

The coefficient 2 before the square root function, \(2\sqrt{x-2}\), represents a vertical stretch. This means that the graph will stretch vertically by a factor of 2. Essentially, the y-values of the original function \(\sqrt{x-2}\) are multiplied by 2.

Step 4: Apply Vertical Shift

Finally, the term \(-1\) at the end of the function, \(2\sqrt{x-2}-1\), represents a vertical shift. Move the entire graph 1 unit down. Consequently, the starting point after the vertical stretch, which was at \((2,0)\), will now be at \((2,-1)\).

Step 5: Sketch the Transformed Graph

Plot the starting point \((2,-1)\) on the graph. Draw the transformed graph starting from this point, following the shape of the stretched and shifted square root function. The graph should curve upwards and to the right, similar to the shape of \(y=\sqrt{x}\), but begin at \((2,-1)\) and rise faster due to the vertical stretch.

Key Concepts

Parent Function

The concept of a parent function is fundamental to understanding graph transformations. A parent function acts as the original or base function from which transformations are derived. In our exercise, the parent function is the square root function, represented by \( y = \sqrt{x} \). Its graph is distinctive, starting at the origin \(0,0\) and rising slowly, curving upwards to the right in a smooth, gentle arc.
Understanding the shape and properties of the parent function helps in predicting how transformations like shifts and stretches will alter the graph. With our square root parent function, any transformation will primarily translate or reshape this characteristic arc that increases to the right.

Horizontal Shift

Horizontal shifts involve moving the graph of a function left or right on the coordinate plane. This is often dictated by changes inside the function's argument, such as \( x - c \). In our exercise, we have \( x-2 \), which indicates a shift to the right.
The term \( x-2 \) means that each x-value of the parent function shifts 2 units to the right. This is because the function now achieves the same outputs as it did previously, 2 steps later on the x-axis.
Starting from the original point \(0,0\) of the parent function \( y=\sqrt{x} \), this rightward movement takes us to a new starting point at \(2,0\). The entire graph follows this horizontal transformation, maintaining its overall shape while shifting position.

Vertical Stretch

Vertical stretches modify a graph by multiplying all its y-values by a certain factor. This transformation results in the graph becoming "taller" or "thinner." In the context of our exercise, the vertical stretch is seen in the factor of 2 that multiplies the square root function, shown as \( 2\sqrt{x-2} \).
This vertical stretch means that any point on the graph will have its y-coordinate multiplied by 2. Consequently, the graph not only starts at the new point after the horizontal shift but also rises more steeply, doubling the height of any point compared to the parent function \( \sqrt{x} \).
This change results in a graph that is both shifted and more exaggerated vertically, still maintaining the overall shape but stretching away from the x-axis more than the original.

Vertical Shift

A vertical shift moves the entire graph up or down on the coordinate plane without altering its shape. In our exercise, this is encapsulated by the term \(-1\) in \( 2\sqrt{x-2} - 1 \), indicating a shift downward by 1 unit.

• The effect of this transformation is simple: after stretching the graph vertically, every y-value is decreased by 1.
• This moves the starting point from \(2,0\) down to \(2,-1\).

The curve remains identical in shape and stretch, but every point has its y-coordinate decreased by 1. This slight downward adjustment gives the final position of our transformed graph, balancing the previous transformations effectively.