Problem 19 - 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. $$ g(x)=\frac{1}{2} \sqrt{x+2} $$

Step-by-Step Solution

Answer

The graph of the function \(g(x)=\frac{1}{2} \sqrt{x+2}\) is obtained by shifting the graph of the function \(f(x)=\sqrt{x}\) 2 units to the left and scaling vertically by a factor of \(\frac{1}{2}\).

Step 1: Graph the original function

Begin by graphing the parent function \(f(x)=\sqrt{x}\). The graph should start at the origin and increase slowly, since for every \(x\), \(f(x)\) is the square root of \(x\) which gets bigger more slowly than \(x\) itself.

Step 2: Apply the horizontal shift

Now apply a horizontal shift to the left by 2 units. This corresponds to the \(+2\) inside the square root in \(g(x)\). This means that the entire graph of \(f(x)\) should be moved 2 units to the left. As a result, instead of the graph starting at the origin, it should now start at \((-2,0)\).

Step 3: Apply the vertical scaling

Lastly, apply a vertical scaling to the graph. This corresponds to the \(\frac{1}{2}\) multiplier in \(g(x)\). This means every y-value should be halved. So if on \(f(x)\), when \(x=1\), \(y=1\), on \(g(x)\) when \(x=1\), \(y = \frac{1}{2}\).

Step 4: Draw the new graph

Now, draw the new graph incorporating both transformations. Remember, the graph should start at \(-2, 0\) rather than the origin because of the horizontal shift, and it should rise more slowly than the graph of \(f(x)=\sqrt{x}\) because of the vertical scaling due to the \(\frac{1}{2}\) multiplier.

Key Concepts

Square Root Function

The square root function is fundamental in mathematics, represented as \( f(x) = \sqrt{x} \). Let's explore its characteristics to understand why it's essential for transformations. This function starts at the origin, \((0,0)\), and progresses slowly upwards. Unlike linear functions where y-values increase consistently with x, here each increase in x results in smaller increases in y. For example:

• When \( x = 0 \): \( f(x) = 0 \)
• When \( x = 1 \): \( f(x) = 1 \)
• When \( x = 4 \): \( f(x) = 2 \)

The graph is a gentle curve, starting at the origin and rising to the right. It's crucial to understand this original graph since transformations modify it to our new function, \( g(x) = \frac{1}{2} \sqrt{x+2} \). By graphing \( f(x) \) first, we set the stage for applying both horizontal shifts and vertical scaling.

Horizontal Shift

A horizontal shift adjusts the position of the graph sideways. In our exercise, the function \( g(x) = \frac{1}{2} \sqrt{x+2} \) involves a horizontal shift due to the \(+2\) inside the square root. Here's how it works:

• The expression \( +2 \) means we shift the graph 2 units to the left. It's counterintuitive since one might expect a positive sign to move in the positive direction, but within functions, adding a number inside the argument (inside the square root for this example) moves the graph in the opposite direction.
• This transformation changes the starting point from the origin (\(0,0\)) to \((-2,0\)).

Visualizing the shift: the whole curve slides left, but its shape remains unchanged. This shift sets a new baseline for any additional transformations, like vertical scaling.

Vertical Scaling

Vertical scaling adjusts how much the graph stretches or compresses vertically. In \( g(x) = \frac{1}{2} \sqrt{x+2} \), the factor \( \frac{1}{2} \) indicates scaling. Understanding vertical scaling:

• The factor \( \frac{1}{2} \) dictates that each y-value in the parent function \( f(x) = \sqrt{x} \) is halved. If \( y \) was originally 2, it will now be \( 1 \).
• This modification influences how steep or flat the graph appears. Each point is moved closer to the x-axis, resulting in a graph that rises less steeply than the original.

The purpose of vertical scaling is often to adjust the range or fit of the graph to better model specific data or equations. Here, combining this with a horizontal shift, we graph the transformed function with its altered appearance, showing a comprehensive view of how transformations interact together.