Problem 18 - Algebra and Trigonometry

Question

\(15-18\) Explain how the graph of \(g\) is obtained from the graph of \(f .\) $$ \begin{array}{ll}{\text { (a) } f(x)=\sqrt{x},} & {g(x)=-\sqrt{x}+1} \\\ {\text { (b) } f(x)=\sqrt{x},} & {g(x)=\sqrt{-x}+1}\end{array} $$

Step-by-Step Solution

Answer

(a) Reflect \( f(x) \) over the x-axis, then shift up 1; (b) Reflect \( f(x) \) over the y-axis, then shift up 1.

Step 1: Identify the Base Graph

The base function given is \( f(x) = \sqrt{x} \). This is a square root function, which is defined for \( x \geq 0 \) and has a graph that starts at the origin (0,0) and increases to the right, forming a curve in the first quadrant.

Step 2 (a): Analyze the Transformation for g(x)

For \( g(x) = -\sqrt{x} + 1 \), perform the following transformations on the graph of \( f(x) = \sqrt{x} \):- Reflect the graph of \( f(x) \) across the x-axis. This is due to the negative sign in front of \( \sqrt{x} \), which changes the graph from a curve increasing upwards to a curve decreasing downwards.- Translate the graph upward by 1 unit. This is done by adding 1 to the function, shifting the entire graph up by 1.

Step 3 (b): Analyze the Transformation for g(x) with Inverse

For \( g(x) = \sqrt{-x} + 1 \), the transformation steps are:- Reflect the graph of \( f(x) = \sqrt{x} \) across the y-axis. The \( -x \) inside the square root flips the graph horizontally, placing it in the second quadrant.- Translate the graph upward by 1 unit. Adding 1 to the function raises the graph by 1 along the y-axis.

Key Concepts

Square Root Function

The square root function is one of the fundamental functions you will encounter when studying algebra and pre-calculus. The function is typically denoted by \( f(x) = \sqrt{x} \). It is defined only for non-negative values of \( x \) (that is, \( x \geq 0 \)), because the square root of a negative number is not a real number.
The graph of the square root function begins at the origin, which is the point \((0, 0)\), and it forms a gentle upward curve that lies in the first quadrant of the Cartesian plane.

• This curve represents all the points \((x, \sqrt{x})\), where \( x \) is non-negative.
• The shape of the graph is a half-parabola that opens to the right along the x-axis.

This behavior of the graph helps to visualize the concept of the square root: for every positive \( x \), the function provides the number that, when squared, gives the original \( x \). The simplicity and smooth rise of the curve mirror the increasing nature of square roots over these non-negative values.

Reflection Across Axes

Reflections involve flipping the graph over a certain axis, changing its orientation while maintaining its shape. Let's explore two common reflections:
- **Reflection across the x-axis**: This occurs when you multiply the function by -1, i.e., \( g(x) = -f(x) \). For example, the transformation \( g(x) = -\sqrt{x} \) reflects the graph of \( \sqrt{x} \) over the x-axis. The graph, which originally curves upwards, now curves downwards, still beginning at the origin but moving into a downward curve as it progresses right.
- **Reflection across the y-axis**: This occurs when you replace \( x \) with \( -x \) inside the function, i.e., \( g(x) = f(-x) \). For the function \( g(x) = \sqrt{-x} \), the graph of \( \sqrt{x} \) is reflected horizontally. This means the curve that was in the first quadrant now lies in the second quadrant. It's as if the graph were flipped like a book page.
Reflections are useful when analyzing symmetry or mirroring behaviors in functions. They demonstrate how sign changes directly influence the graph's orientation, providing visual insight into how algebraic manipulations can affect graphical output.

Graph Translation

Translation shifts the entire graph of a function up, down, left, or right without altering its shape or orientation. This can be visualized as moving the graph along the plane. For the given transformations:
- **Vertical Translation**: Adding a constant to the function, such as \( g(x) = f(x) + 1 \), results in the graph moving up by that constant. Thus, the graph of \( -\sqrt{x} + 1 \) or \( \sqrt{-x} + 1 \) moves the original square root function's graph up by 1 unit.

• This means every point \((x, y)\) on the original graph shifts to \((x, y+1)\).

- **Horizontal Translation**, not directly mentioned in the exercise solution but also important, involves changes inside the square root. Had we added or subtracted a constant within \( f(x) = \sqrt{x} \), it would shift the graph left or right.
Translations are straightforward yet powerful ways to 'move' functions across axes. They allow us to explore how the graph changes its position while still preserving the fundamental shape and symmetry of the square root function.