Problem 20 - 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. $$ g(x)=2 \sqrt{x+1} $$

Step-by-Step Solution

Answer

The graph of \(g(x) = 2 \sqrt{x+1}\) can be obtained by vertically stretching the graph of \(f(x) = \sqrt{x}\) by a factor of 2, and then shifting it one unit to the left.

Step 1: Graph the basic function

Start by graphing the basic function \(f(x) = \sqrt{x}\). This is a function that starts at the origin (0, 0) and increases as x increases, forming a curve.

Step 2: Identify the transformations

Looking at the function \(g(x) = 2 \sqrt{x+1}\), there are two transformations to the basic square root function. The '2' before the square root is a vertical stretch: this means the y-values of the function are doubled. The '+1' inside the square root is a horizontal shift: it moves the function one unit to the left.

Step 3: Apply the transformations

Now apply these transformations to the graph of \(f(x) = \sqrt{x}\). All the y-values will be double of what they were in the original function, and the entire curve will shift one unit to the left. This forms the graph of \(g(x) = 2 \sqrt{x+1}\).

Key Concepts

Square Root Function

The square root function is an essential building block for understanding many mathematical concepts. It is defined as \( f(x) = \sqrt{x} \). In simpler terms, this function returns the square root of a given non-negative input \( x \). The graph of this function is quite distinctive:

• It starts at the origin, (0, 0), which means it doesn't output a value for negative \( x \).
• As \( x \) increases, \( f(x) \) also increases, creating a curve that rises to the right.
• The curve's slope decreases as \( x \) becomes larger, meaning it flattens out but never truly becomes horizontal.

Understanding this basic function is crucial before applying any transformations such as stretches or shifts. These transformations alter its shape and position but the underlying process of taking the square root remains the same.

Vertical Stretch

When we discuss a vertical stretch, we focus on changes in the y-values of a function's graph. The expression \( 2\sqrt{x} \) implies that each y-value of the basic square root function \( \sqrt{x} \) is now doubled. Here's what a vertical stretch involves:

• All points on the graph are moved further away from the x-axis. So, if a point on \( f(x) = \sqrt{x} \) was \( (a, b) \), in \( 2\sqrt{x} \) it becomes \( (a, 2b) \).
• This makes the graph appear stretched vertically, making it steeper than the parent curve.
• The overall shape of the curve remains the same; however, its rate of increase is exaggerated.

A vertical stretch impacts how quickly the function's value grows as \( x \) increases, making it a crucial transformation for graph manipulation.

Horizontal Shift

Horizontal shifts cause the graph of a function to move left or right. In the function \( g(x) = 2\sqrt{x+1} \), the \( +1 \) inside the square root indicates a horizontal shift. Here is how it works:

• The addition of \(+1\) causes the entire graph to shift one unit to the left. This is because you must subtract 1 from \( x \) to 'compensate', which mirrors the graph relative to the y-axis.
• Every point on the basic graph \( \sqrt{x} \) shifts; so a point originally at \( (a, b) \) will move to \( (a-1, b) \).
• The shape of the graph remains unchanged, but its position on the x-axis adjusts.

These shifts help in aligning graphs correctly along the x-axis without altering the fundamental nature of their increase or decrease. It’s a vital tool for plotting precise graph positions in coordinate plane transformations.