Problem 22 - College Algebra

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

Firstly, \(f(x) = \sqrt{x}\) should be graphed which starts at the origin and moves to the right. For graphing \(h(x) = \sqrt{x+1}-1\), we start the graph at (-1,0) because of \(x+1\) (shifting the function to left by 1 unit). Then, because of -1 outside of the function, shift everything down by one unit. Therefore, \(h(x)\) equals to \(f(x)\) that is shifted one unit to the left and one unit down.

Step 1: Graphing the base function

Start by drawing the square root function \(f(x) = \sqrt{x}\). This is a basic function and it begins at the origin (0,0) and increases slowly to the right. The graph will never go to the left of the y-axis because taking square root of negative number is undefined in real numbers.

Step 2: Moving the function to the left

In \(h(x) = \sqrt{x+1}-1\), \(x+1\) is inside the square root. This indicates a shift to the left, not to the right. The function \(h(x)\) will start from the point (-1,0) not from (0,0) because if \(x = -1\), the inside of the root becomes zero and hence the entire function also becomes 0. So, graph the function with this new starting point.

Step 3: Moving the function down

Notice there's a -1 outside the square root in \(h(x) = \sqrt{x+1}-1\). This means the function has been moved down by 1 unit. So, take your graph from Step 2 and shift every point down by one unit to correctly depict \(h(x)\).

Step 4: Final graph of \(h(x)\)

After moving to the left by 1 and down by 1. The graph has both the shape of the square root function and the correct transformations as indicated by \(h(x) = \sqrt{x+1}-1\).

Key Concepts

Function Transformations

Function transformations are an essential concept in understanding how to manipulate and move the graphs of functions without changing their shapes. Imagine you have a piece of transparent paper with a graph on it. When you slide, flip, stretch, or compress this paper, you are applying transformations to the function’s graph.

For the square root function, transformations can be applied in the following ways:

• Vertical Translations: Adding or subtracting a number on the outside of the square root shifts the graph up or down respectively.
• Horizontal Translations: Adding or subtracting a number within the square root shifts the graph left or right respectively.
• Vertical Stretch/Compression: Multiplying the square root by a number greater than 1 stretches the graph away from the x-axis, while a number between 0 and 1 compresses it towards the axis.
• Reflection: Multiplying the function by -1 reflects it over the x-axis.

Understanding these transformations allows us to manipulate the function and predict how the graph will change.

Graph Translations

Graph translations refer to the shifting of the entire graph of a function in a specific direction. In the context of graphing the given square root function, identifying the graph translations is crucial for accurate placement of the graph.

For our exercise example, the translation involves two movements:

• Left by 1 unit: The presence of the +1 inside the square root, within the function \(h(x) = \sqrt{x+1}-1\), translates the graph 1 unit to the left. Remember, if the addition is occurring inside the square root, the graph moves opposite to the direction you might initially think.
• Down by 1 unit: The -1 outside the square root indicates a downward shift of the entire graph by 1 unit.

The combined effect of these translations is a new starting point for the reflected graph at the coordinate (-1, -1).

Square Root Graph Properties

Square root functions have distinctive properties that define their graphs. First and foremost, the square root graph is known as a radical graph and has a characteristic 'half-parabola' shape opening towards the right.

Here are some primary properties of square root graphs:

• Domain: The set of all possible x values. For the base function \(f(x) = \sqrt{x}\), the domain is [0, +∞) as square roots of negative numbers are not real.
• Range: The set of all possible y values. For \(f(x)\), the range is also [0, +∞).
• Starting Point: For \(f(x)\), the graph starts at the origin (0,0), but this can change with translations.
• Asymptotic Behavior: Although not typically considered an asymptote, the y-axis acts as a boundary for the graph since it cannot extend into negative x-values.
• Increasing Nature: Square root functions always increase, but the rate of increase slows down as x grows larger.

These properties are essential for sketching a square root graph, and they become even more helpful when functions involve transformations, as they serve as a reference point for the changes that occur.