Problem 78 - Precalculus Essentials

Question

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=\sqrt{x+1}-1$$

Step-by-Step Solution

Answer

The graph of \(h(x)=\sqrt{x+1}-1\) can be obtained by first plotting the square root base function, then shifting the graph 1 unit to the left (due to \(+1\) within the square root) and afterwards, shifting it 1 unit downwards (due to the \(-1\)).

Step 1: Plot Basic Function

Start by plotting the basic function \(f(x)=\sqrt{x}\). Choose a range of \(x\) values, calculate the respective \(y=f(x)\) for all values and plot these points on your graph. Some key points to include can be \(x=0\) and \(x=1\).

Step 2: Apply Horizontal Shift

The term \(x+1\) under the square root sign indicates a horizontal shift to the LEFT by 1 unit. This means every point on the \(f(x)\) graph will move 1 unit to the left to form the graph of \(h(x)\).

Step 3: Apply Vertical Shift

The term \(-1\) in the function \(h(x)=\sqrt{x+1}-1\) indicates a downward shift by 1 unit. So, after applying the horizontal shift, shift every point on the graph down by 1 unit.

Step 4: Obtain the Graph of h(x)

After applying both shifts to the original points on the graph of \(f(x)=\sqrt{x}\), plot the corresponding points to get the graph of \(h(x)=\sqrt{x+1}-1\). Connect these points to visualize the function.

Key Concepts

Square Root Function

The square root function is one of the most basic yet important functions in mathematics. It is defined as: \(f(x) = \sqrt{x}\). This function only takes non-negative values of x, meaning it starts from x = 0 and increases gradually as x increases.

The standard graph of the square root function, \(f(x) = \sqrt{x}\), is a curve that begins at the origin (0,0) and rises to the right, never crossing the x-axis. It forms a gentle curve because as x gets larger, the function's growth slows down.

- **Key points to plot**:
- When x = 0, y = 0
- When x = 1, y = 1
- Increasing with a decreasing rate

This function is the starting point before applying any transformations like shifts.

Horizontal Shift

In function transformations, a horizontal shift occurs when the graph of a function moves left or right. For the square root function, the equation looks like \(f(x) = \sqrt{x + c}\).

- **Understanding the shift**:
- If c is positive, shift the graph to the left by c units (as seen in \(\sqrt{x+1}\))
- If c is negative, the graph shifts to the right by \(|c|\) units

In our case, \(\sqrt{x+1}\) tells us to shift every point of the original square root function one unit to the left.
This moves the starting point from (0,0) to (-1,0), effectively beginning the graph's curve at this new point.

Vertical Shift

A vertical shift moves a graph up or down without changing its shape. In a transformation equation, such as \(g(x) = f(x) + k\), the function shifts vertically.

- **For positive k**:
- The graph shifts upward by k units
- **For negative k**:
- The graph shifts downward by |k| units

In \(h(x)=\sqrt{x+1} - 1\), the term \(-1\) indicates a shift down by 1 unit. This means every point on the graph of \(\sqrt{x+1}\) is moved one unit down, adjusting the entire curve downwards.

Function Transformation

Function transformations are powerful tools that allow us to manipulate and visualize different versions of standard functions.

By understanding how transformations like horizontal and vertical shifts work, we can predict and graph changes in more complex equations. Graphing transformations typically involve:

• **Horizontal shifts**: alterations along the x-axis
• **Vertical shifts**: adjustments up or down along the y-axis

To transform the square root function \(f(x)=\sqrt{x}\) into \(h(x)=\sqrt{x+1}-1\), simply:
- Shift the graph of \(f(x)\) 1 unit to the left
- Then, move it 1 unit downward

This creates a new graph of \(h(x)\) that models the transformed function with clear visual changes apparent from the starting function.