Problem 91 - Calculus Volume 1

Question

For the following exercises, use the graph of \(f(x)=\sqrt{x}\) to graph each transformed function \(g\) . $$ g(x)=-\sqrt{x}-1 $$

Step-by-Step Solution

Answer

The graph of \( g(x) = -\sqrt{x} - 1 \) is the reflection of \( \sqrt{x} \) over the x-axis, shifted down by 1 unit.

Step 1: Identify the Parent Function

The parent function is given as \( f(x) = \sqrt{x} \). This function represents a square root graph that starts at the origin \((0, 0)\) and increases gradually to the right.

Step 2: Analyze the Transformation of g(x)

The transformed function is \( g(x) = -\sqrt{x} - 1 \). Here, the transformation operations include reflection, vertical translation, and horizontal translation.

Step 3: Reflect Over the x-axis

The negative sign in front of the square root, \( -\sqrt{x} \), indicates a reflection of the graph of \( \sqrt{x} \) over the x-axis. This flips the graph upside down.

Step 4: Translate Vertically Downward

The term \(-1\) adjusts the graph downward by 1 unit. As a result, each point on the graph \(-\sqrt{x} \) is shifted down by 1 unit.

Step 5: Plot Key Points and Sketch the Graph

Due to the transformations: The point \((0, 0)\) on \( f(x) \) changes to \((0, -1)\) on \( g(x) \). Similarly, \((1, -1)\) becomes \((1, -2)\), and \((4, -2)\) becomes \((4, -3)\). Using these transformed points, sketch the transformed graph, ensuring it is reflected and shifted accurately.

Key Concepts

Parent Function

A parent function is the simplest form of a given type of function. In this exercise, our parent function is \(f(x) = \sqrt{x}\). This function is the "parent" of square root functions because every other square root function is a transformation of it. The graph of \(\sqrt{x}\) starts at the origin \((0, 0)\) and gradually rises to the right. This rising curve is the hallmark of the square root function. Parent functions are important because they act as the base graph that we modify to show transformations.

Reflection Over x-axis

Reflection over the x-axis is like flipping a graph upside down. When we have a function like \(-\sqrt{x}\), the negative sign in front of the square root indicates that we need to reflect the entire graph over the x-axis. This means that all the positive y-values of the original square root function \(\sqrt{x}\) turn into negative y-values for \(-\sqrt{x}\). It is as if a mirror were placed on the x-axis, and the graph flips accordingly. This reflection fundamentally changes the direction in which the graph opens.

Vertical Translation

Vertical translation involves moving the graph up or down without altering its shape. In \(g(x) = -\sqrt{x} - 1\), the "-1" at the end tells us to shift the entire graph one unit downward. Each point on the graph of \(-\sqrt{x}\) moves straight down by one unit. This affects how low the graph reaches, but it maintains the overall shape and orientation after the reflection. Vertical translations can make the graph appear as if it started and ended at different vertical positions on the y-axis.

Graph Sketching

Graph sketching involves plotting the key points to visualize the transformation. After applying the transformations, we can sketch \(g(x) = -\sqrt{x} - 1\). For instance:

• The original point \((0, 0)\) from \(f(x)\) becomes \( (0, -1) \) on \(g(x)\), due to the reflection and downward shift.
• The point \((1, 1)\) moves to \((1, -2)\).
• The point \((4, 2)\) becomes \((4, -3)\).

By plotting these key points and connecting them smoothly, the graph reflects both transformations correctly. It's essential to sketch lightly, allowing adjustments, as visual accuracy is key for understanding transformations.