Problem 74 - 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. $$h(x)=\sqrt{-x+1}$$

Step-by-Step Solution

Answer

The graph of \(h(x)=\sqrt{-x+1}\) is a reflected version of the graph of \(f(x)=\sqrt{x}\) over the y-axis, followed by a shift to the right by one unit.

Step 1: Graph the Original Function

Firstly, graph the function \(f(x)=\sqrt{x}\). This function starts at the origin (0,0) and increases slowly, representing the square root of each x-value.

Step 2: Determine the Transformation

Look at the function \(h(x)=\sqrt{-x+1}\). Compared with the basic form, we can see that it is derived from the basic function through two transformations: 1) The x is negated, which represents a reflection about the y-axis. This means flipping the original square root function over the y-axis. 2) There is a positive 1 added after the x, which represents a horizontal shift to the right by 1 unit.

Step 3: Apply the Transformation and create the new graph

Apply the determined transformations to the original graph of \(f(x)=\sqrt{x}\). First, reflect the original graph over the y-axis. Next, shift the reflected graph to the right by one unit to get the graph of \(h(x)=\sqrt{-x+1}\).

Key Concepts

Square Root Function

The square root function, denoted as \(f(x) = \sqrt{x}\), is one of the fundamental mathematical functions used to describe the relationship where a number, \(x\), is squared to produce some result. When graphed, this function has some specific characteristics which are easy to recognize.

• The graph starts at the origin, point (0,0), since the square root of 0 is 0.
• The curve extends to the right, increasing as \(x\) increases, but at a decreasing rate.
• This graph is always in the first quadrant, because both \(x\) and \(y\) values are non-negative.
• The square root function produces half of a parabola lying on its side.

Understanding the shape and characteristics of the square root function helps in visualizing and applying graph transformations effectively.

Reflection

Reflection is a transformation that flips the graph of a function over a specific line, creating a mirror image. For the problem at hand, you'll deal with a reflection across the y-axis. Here's how it works:

• For the function \(h(x)=\sqrt{-x+1}\), the negative sign before \(x\) indicates reflection over the y-axis.
• This means that every point (x, y) on the original \(f(x) = \sqrt{x}\) will be transformed to (-x, y).
• This flipping action causes the graph to appear as if it’s in reflection mode, shifting portions of the curve into the opposite quadrant.

It's crucial to follow this step accurately since the reflection significantly alters how the function appears in its visual form.

Horizontal Shift

Horizontal shifts are used to move the graph of a function to the left or right without altering its shape. This transformation is indicated by added or subtracted values inside the function’s argument.
In the function \(h(x) = \sqrt{-x+1}\), observe:

• The term \(+1\) inside the square root implies that there is a horizontal shift.
• Generally, the graph of \(\sqrt{-x}\) would begin at the origin; however, adding \+1\ moves every point of the graph to the right by 1 unit. This is counter-intuitive because we modify the term directly associated with x.
• Thus, any point (x, y) on the original graph shifts to (x+1, y) on the transformed graph.

By mastering these concepts, you'll be able to visually adjust a function to its appropriate position on a graph.