Problem 28 - A Graphical Approach to Precalculus with Limits

Question

The graph of \(y=-6 \sqrt{x}\) can be obtained from the graph of \(y=\sqrt{x}\) by vertically stretching by applying a factor of __________ and reflecting across the __________-axis.

Step-by-Step Solution

Answer

Vertically stretch by a factor of 6 and reflect across the x-axis.

Step 1: Identify the Ordinary Graph

The given problem involves the manipulation of the graph of the basic function \(y = \sqrt{x}\). This basic function represents a curve that starts at the origin and extends in a positive direction along the x-axis, forming the right half of a sideways parabola.

Step 2: Analyze the New Graph Equation

The new equation given is \(y = -6\sqrt{x}\). This equation involves two transformations applied to the basic function \(y = \sqrt{x}\). Understanding these transformations will explain how the basic graph is altered.

Step 3: Determine the Vertical Stretch

Looking at the equation \(y = -6\sqrt{x}\), the coefficient in front of \(\sqrt{x}\) is \(-6\). The absolute value of \(-6\) is 6, which signifies a vertical stretch of the graph by a factor of 6.

Step 4: Determine Reflection Across Axis

Since there is a negative sign before the coefficient \(-6\sqrt{x}\), this indicates a reflection across the x-axis. The graph of \(y = \sqrt{x}\) will be reflected downwards, converting positive outputs to negative values.

Key Concepts

Vertical Stretch

A vertical stretch is a transformation that changes the shape of a graph by pulling it away or pushing it towards the x-axis. This effect is achieved by multiplying the function by a constant factor.
For the square root function, like with our example, we see the basic function \(y = \sqrt{x}\) being modified to \(y = -6\sqrt{x}\). Here, the key component is the multiplier, \(6\), which indicates how much the graph stretches.
This vertical stretch can be understood as making the graph taller or more prominent. When the absolute value of the factor is greater than 1, the transformation makes the graph appear taller and steeper.
To visualize it, imagine applying a rubber band horizontally on a tabletop and then pulling the table away, which stretches the graph upwards, creating a narrower appearance.

Reflection

Reflection in graphs is like seeing a mirror image. When a reflection transformation is applied, it flips the graph over a designated axis. In this exercise, that axis is the x-axis. For the function \(y = -6\sqrt{x}\), the negative sign before the coefficient indicates this flip.
In simple terms, every point above the x-axis is mirrored below it, and vice versa.
This transformation changes positive outputs (upward y-values of \(y = \sqrt{x}\)) to negative outputs in \(y = -6\sqrt{x}\). As a result, instead of the graph rising upwards as \(x\) increases, it falls below the x-axis, forming a mirrored version of the initial graph.
This concept of reflection is crucial as it alters the direction in which the graph extends.

Square Root Function

The square root function, represented as \(y = \sqrt{x}\), starts at the origin (0,0) and only operates in a positive direction along the x-axis. This makes the graph appear as half of a sideways parabola.
It increases steadily as \(x\) grows, but its rate of increase slows down, giving it a gentle curve upward.
The function is defined only for non-negative \(x\)-values because the square root of a negative number is not real in basic algebra.
In transformations, this basic shape can be stretched, compressed, or reflected. For example, when considering \(y = -6\sqrt{x}\), the function undergoes a vertical stretch and reflection as discussed earlier, drastically changing its visual representation while following the core principle of the square root function.