Problem 73 - 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. $$h(x)=\sqrt{-x+2}$$

Step-by-Step Solution

Answer

The graph of function \(h(x)=\sqrt{-x+2}\) is the graph of the basic square root function, \(f(x)=\sqrt{x}\) reflected across the y-axis and then shifted 2 units to the right.

Step 1: Graphing the basic function

Graph the basic square root function, \(f(x)=\sqrt{x}\). When \(x=0\), \(y=f(x)=0\), for \(x=1\), \(y=1\), and for \(x=4\), \(y=2\). Plotting these points and drawing the curve would give a basic idea of the square root function.

Step 2: Reflection about y-axis

The negative sign indicates a reflection of the graph about the y-axis. So, reflect all points of the basic graph about the y-axis. The reflection of a point \(P(x, y)\) about y-axis is \(P’(-x, y)\).

Step 3: Horizontal Shift

The '+2' in the brackets indicates a horizontal shift to the right by 2 units. After reflection, shift all the points 2 units to right.

Step 4: Plotting the final graph

Upon completing the above transformations, plot the final graph. The final graph should maintain the basic shape of square root function after transformations.

Key Concepts

Square Root Function

The square root function is one of the basic mathematical functions you’ll encounter. It is denoted as \( f(x) = \sqrt{x} \), and its graph has a unique shape known as the "square root curve." This function only takes non-negative values of \( x\), because square roots of negative numbers are not defined in the set of real numbers.
The basic shape of the square root function resembles a gradual upward curve starting from the origin (0, 0), where both the x-coordinate and y-coordinate are zero. As you increase \( x \), \( y \) also increases, but at a slower rate due to the nature of square roots. Some key points on this graph include:

• \( (0, 0) \) where \( x = 0 \)
• \( (1, 1) \) as \( \, \sqrt{1} = 1 \)
• \( (4, 2) \) since \( \, \sqrt{4} = 2 \)

This graph helps in visualizing how the function grows as \( x \) increases. Understanding the structure of this basic function is crucial before applying any transformations.

Reflection across the y-axis

Reflection across the y-axis is a type of transformation used to modify the appearance and orientation of functions on a graph. In the context of the square root function, it involves flipping the graph of the function \( f(x) = \sqrt{x} \) to form \( f(-x) = \sqrt{-x} \).
To perform this transformation, you follow the rule \( P(x, y) \rightarrow P'(-x, y) \). This means that you change the sign of the x-coordinate for each point while keeping the y-coordinate the same. Reflection effectively produces a mirror image of the graph on the opposite side of the y-axis.
For example, in the function \( h(x)=\sqrt{-x+2} \), the negative sign inside the square root indicates this reflection process. The points you might previously have on the right side of the y-axis are now interpreted on the left side, which turns the initial trajectory of the square root function in the opposite horizontal direction.

Horizontal Shift

A horizontal shift involves moving the entire graph of a function either left or right. It is different from simply moving it up or down, and it is particularly common with functions like polynomials or square roots.
For a function \( f(x) = \sqrt{x} \), if you have an expression like \( f(x+c) \), this indicates a horizontal shift. But notice: if \( c > 0 \), the graph shifts \( c \) units to the left, due to the opposite nature of horizontal shifts; if \( c < 0 \), then to the right.
In the function \( h(x) = \sqrt{-x + 2} \), after reflecting over the y-axis from \( \sqrt{-x} \), you use the "+2" inside the parenthesis to shift the graph 2 units to the right. This adjustment allows the graph to reach a new position, making sure the transformation retains the correct logical sequence and retains the function's shape even though it has shifted location.