Problem 31 - Algebra 2

Question

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt{9 x-9}\)

Step-by-Step Solution

Answer

The function \(y=\sqrt{9 x-9}\) can be rewritten as \(y=\sqrt{9 (x-1)}\). It is a transformation of the parent function \(y=\sqrt{x}\) with a horizontal shrink by a factor of 1/9 and a shift one unit to the right. The graph starts from (1,0) and increases more quickly towards the y-axis as x increases.

Step 1: Identify the parent function and the parameters

Here, the parent function is \(y=\sqrt{x}\). The given function is \(y=\sqrt{9x-9}\). We can rewrite this as \(y=\sqrt{9(x-1)}\) which is now in the form \(y=a\sqrt{b(x-h)}+k\). In this case, a=1, b=9, h=1 and k=0.

Step 2: Determine the transformations

The value of a=1 means there is no vertical stretch. Because b>1, we have a horizontal shrink or compression by a factor of 1/9. The positive value of h=1 means the graph is shifted one unit to the right. Since k=0, there is no vertical shift.

Step 3: Describe the graph

The graph of \(y=\sqrt{x}\) starts at (0,0) and increases as it moves to the right. Using the transformations calculated, we know that our graph will start at (1,0) due to the horizontal shift. Then, instead of spreading out as the function increases, it will move towards the y-axis more quickly because of the horizontal compression (shrink).

Key Concepts

Parent Function

In graphing transformations, the parent function serves as the basic building block before any transformations are applied. For the given exercise, the parent function is the square root function \( y = \sqrt{x} \). This is an important concept because it provides the starting point for understanding how transformations will modify the graph.

• The graph of \( y = \sqrt{x} \) starts at the origin, point (0, 0), and rises gradually to the right.
• Understanding the parent function helps in visualizing changes when parameters are altered.

Using the parent function, any changes to the graph can be understood as deviations from this original shape through transformations like shifts, stretches, and compressions.

Square Root Function

The square root function, \( y = \sqrt{x} \), is characterized by its unique shape. It begins at a starting point and rises gradually to the right. This function is smooth and increases without bound as \( x \) increases.

• The graph rises slowly initially and speeds up as \( x \) gets larger.
• This function does not exist for negative values of \( x \) since the square root of a negative number is not real.

Understanding the characteristics of the square root function is essential for applying graph transformations effectively, such as evaluating how shifts and compressions affect its overall shape and position.

Horizontal Compression

Horizontal compression occurs when the x-coordinates are "squeezed" towards the y-axis. In the context of the function \( y=\sqrt{9(x-1)} \), the parameter \( b=9 \) indicates a horizontal compression.

• This compression is by a factor of \( \frac{1}{b} \), which in this exercise, means \( \frac{1}{9} \).
• The effect of this transformation is that the graph of the square root function becomes narrower as it rises.

Horizontal compression makes the function react faster to changes in \( x \) compared to the parent function, leading to a steeper climb up the y-axis.

Horizontal Shift

Horizontal shifts move the graph left or right on the coordinate plane. For the function \( y=\sqrt{9(x-1)} \), the \( h \) value is 1, which means there is a horizontal shift.

• The graph is shifted to the right by 1 unit because \( h \) is positive.
• This transformation moves the starting point of the parent function from (0, 0) to (1, 0).

This horizontal shift is crucial for understanding where the graph begins and how it's positioned relative to the y-axis, allowing us to place the transformed graph correctly on the coordinate plane.