Problem 15 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{array}{l}f(x)=-\sqrt{x} \\\g(x)=\sqrt{x+1} \\\h(x)=\sqrt{x-2}+1\end{array}$$.

Step-by-Step Solution

Answer

Sketch \(f(x) = -\sqrt{x}\) as an inverted square root graph starting from the origin. Sketch \(g(x) = \sqrt{x+1}\) as a standard square root graph shifted left by one unit. Sketch \(h(x) = \sqrt{x-2}+1\) as a standard square root graph shifted two units to the right and one unit upwards. Combine all three sketches into one graph.

Step 1: Sketching \(f(x) = -\sqrt{x}\)

The function \(f(x) = -\sqrt{x}\) is a reflection of the basic square root function \(\sqrt{x}\) in the x-axis. In other words, it's the basic square root graph flipped upside down. The graph will start from the origin (0,0) and decrease while moving to the right.

Step 2: Sketching \(g(x) = \sqrt{x+1}\)

The function \(g(x) = \sqrt{x+1}\) is a horizontal shift of the base square root function \(\sqrt{x}\). The \(+1\) inside the square root function means the graph is shifted one unit to the left. The graph will start from the point (-1, 0) and increase while moving to the right.

Step 3: Sketching \(h(x) = \sqrt{x-2}+1\)

The function \(h(x) = \sqrt{x-2}+1\) is a combination of horizontal and vertical shift. The -2 inside the square root shifts the graph 2 units to the right, and the +1 outside the square root shifts the graph 1 unit up. The graph will start from (2,1) and increase while moving to the right.

Step 4: Combining the three sketches

Combine the sketches from steps 1-3 into one graph. Make sure to label each graph to avoid confusion.

Key Concepts

Square Root Functions

Square root functions are a type of function represented by a radical symbol, specifically \( f(x) = \sqrt{x} \). These functions depict the relationship of the square root of a variable. The graph of a basic square root function is a curve that starts at the origin (0,0) and increases as it moves to the right. The shape is known as half of a "parabola" lying on its side.

• The function is defined only for non-negative values since square roots of negative numbers are non-real in basic function terms.
• Its domain is \( x \geq 0 \), meaning it only takes zero and positive numbers.
• Its range is also non-negative, starting from zero and extending upwards. \( f(x) \geq 0 \)

Since the square root function's graph is always increasing, each x-value produces a unique y-value, making it an increasing function.

Graph Transformations

Graph transformations involve shifting, reflecting, or stretching a graph in the coordinate system to create new functions. These transformations include translations such as moving the graph left, right, up, or down, reflections, and stretching/compressing the graph.Reflections:- The function \( f(x) = -\sqrt{x} \) reflects the graph of \( \sqrt{x} \) across the x-axis. Thus, the usual upward-curving graph flips to curve downwards.
Horizontal Translations:- The function \( g(x) = \sqrt{x+1} \) exemplifies a leftward horizontal shift. The \( +1 \) inside the square root signals a move one unit to the left, from (0,0) to (-1,0).
Combined Shift:- In \( h(x) = \sqrt{x-2}+1 \), there is a shift both horizontally and vertically. The \( x-2 \) moves the graph 2 units to the right, and the \( +1 \) outside moves it 1 unit up. Through these transformations, you can observe how easily the graph morphs while the essential features of its shape remain recognizable.

Coordinate System

The coordinate system provides a framework for graphically representing functions. It consists of two axes: horizontal (x-axis) and vertical (y-axis), which intersect at the origin (0,0).
When plotting functions such as square root functions, understanding the interplay of x and y coordinates ensures proper sketching.

• The x-axis acts as a reference for horizontal motion: positive to the right and negative to the left.
• The y-axis measures vertical movement: positive numbers climb upwards, while negatives descend.

For the given square root functions:- \( f(x) = -\sqrt{x} \) will be drawn starting at the origin going downward to the right.- \( g(x) = \sqrt{x+1} \) starts at (-1,0)and moves upwards, slightly to the right.- \( h(x) = \sqrt{x-2}+1 \) initiates at (2,1),proceeding upward.By combining each graph onto the coordinate system, you can visualize their similarities and differences while comparing their transformations.