Problem 59 - Algebra 2

Question

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt{\frac{x-1}{4}}-2\)

Step-by-Step Solution

Answer

The function \(y=\sqrt{\frac{x-1}{4}}-2\) transforms the parent function \(y = \sqrt{x}\) by shifting it 1 unit to the right and 2 units down, and compressing it vertically by a factor of 1/4. Its graph starts from point (1,-2) and gradually and slowerly rises as x increases, with no defined y-values for x<1.

Step 1: Identify the Parent Function

The first step is to identify the parent function. In the function \(y=\sqrt{\frac{x-1}{4}}-2\), the parent function is \(y = \sqrt{x}\).

Step 2: Recognize the Transformations

The next step is to interpret the transformations applied to the parent function. The term \(\frac{x-1}{4}\) inside the square root symbol is a shift to the right by 1 unit and a horizontal stretch by a factor of 4. The term -2 at the end of the equation is a vertical shift downwards by 2 units.

Step 3: Rewrite the Function

Now let's rewrite the function in the form of transformations of the parent function. \(y=\sqrt{\frac{x-1}{4}}-2\) can be written as \(y= (\sqrt{x})*\frac{1}{4} + 1 - 2\). This tells that it is a vertically compressed by a factor of 1/4, shifted right by 1 unit, and shifted downward by 2 units.

Step 4: Describe the Graph

Finally, describe the graph of the function. The graph begins at (1, -2), since the function has been shifted 1 unit to the right and 2 units down. It rises gradually and more slowly (due to compression) as x increases. It has no values for x less than 1 since square root of a negative number is undefined in the real number domain.

Key Concepts

Parent Function

In the world of functions and graphs, the "Parent Function" serves as a simple, untransformed function that can help us understand more complex functions.

• The parent function is the simplest form of the family of functions, and modifications are only applied to this form.
• For the exercise at hand, our parent function is \(y = \sqrt{x}\). This represents the most basic form of the square root function.
• This specific function has a starting point at the origin \(0, 0\) and gently curves upwards as \(x\) increases.

Grasping the concept of the parent function is essential because it acts as the baseline before any graph transformations are applied. It's the data's original state, and everything else is a deviation based on this structure.

Horizontal Shift

A "Horizontal Shift" describes how a graph moves left or right on the coordinate plane.

• In our function \( y=\sqrt{\frac{x-1}{4}}-2 \), the horizontal shift occurs due to the term \(x-1\).
• This indicates a shift to the right by 1 unit, because we switch the sign seen within the parenthesis for horizontal shifts. Therefore, whenever you have \(x-c\), it shifts the graph \(c\) units to the right.

Keep in mind:- Horizontal shifts do not affect the function’s shape, only its position.- This is part of analyzing transformations which isolates changes only to the horizontal position of our function on a graph.

Vertical Shift

A "Vertical Shift" involves moving a graph up or down the coordinate plane.

• For the function \( y=\sqrt{\frac{x-1}{4}}-2 \), the vertical shift comes from the \(-2\) term.
• This term tells us to move the graph 2 units down. Contrary to the horizontal shift, the sign is taken directly as is.

Vertical shifts are straightforward since they maintain the shape of the graph. Consistently apply the numerical value of the shift directly to the \(y\)-coordinates.

Square Root Function

The "Square Root Function" is an important fundamental aspect when dealing with graph transformations.

• Its basic form is \(y = \sqrt{x}\).
• Characteristically, it only accepts non-negative values for \(x\), and thus has a domain of \([0, \infty)\).
• It's represented by a curve starting from the origin and rising gently to the right.
• In our complex function, this square root function undergoes transformations that include stretching, compressing, and shifting.

When working with square root functions, always note that any transformations which make the term under the square root negative will make the function undefined in the real number system.