Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=1-\sqrt{x}$$

Step-by-Step Solution

Answer

The graph is a vertically reflected and shifted \(\sqrt{x}\) function.

Step 1: Understand the Basic Function

The basic function we will start with is the square root function, given by \(y = \sqrt{x}\). This is the simplest form of the function that we will transform.

Step 2: Transform the Function with a Vertical Reflection

Next, we transform \(y = \sqrt{x}\) by applying a vertical reflection. This reflection can be achieved by multiplying the function by \(-1\). Thus, we get the function \(y = -\sqrt{x}\). This step reflects the graph of \(y = \sqrt{x}\) across the x-axis.

Step 3: Shift the Function Vertically

Finally, we apply a vertical shift to the function \(y = -\sqrt{x}\). The function we want is \(y = 1 - \sqrt{x}\), so we shift the entire graph of \(y = -\sqrt{x}\) upwards by 1 unit. This is done by adding 1 to the function: \(y = 1 - \sqrt{x}\).

Key Concepts

Vertical Reflection

A vertical reflection is a transformation that flips a graph over a horizontal line, typically the x-axis. When applied to a function, this operation reverses the direction of the graph's peaks and valleys.
For the square root function, starting with the basic form \(y = \sqrt{x}\), multiplying by -1 results in \(y = -\sqrt{x}\).
This action takes every point \((x, y)\) on the graph of \(\sqrt{x}\) and transforms it to \((x, -y)\). This means that if an initial point on \(\sqrt{x}\) is above the x-axis, its reflection will be below the x-axis and vice versa.

Here’s a quick visual: imagine you are looking at a reflection in a pond. The objects above the water will have their reflections below the water. Similarly, the peaks in the graph of \(\sqrt{x}\) become valleys in \(-\sqrt{x}\).
This transformation is crucial in changing the orientation of the graph's shape and is often combined with other transformations like shifts.

Vertical Shift

A vertical shift involves moving the graph of a function up or down along the y-axis. This doesn't change the shape or orientation of the graph but simply translates it vertically. The transformation is carried out by adding or subtracting a constant to the function.
In our case, after applying the vertical reflection to get \(y = -\sqrt{x}\), a vertical shift is used to obtain \(y = 1 - \sqrt{x}\).
Here, adding 1 shifts the entire graph of \(-\sqrt{x}\) upwards by 1 unit.

You can think of it like moving a painting up a wall. The painting remains the same, but its position on the wall changes.

• When a function is given by \(y + c = f(x)\), the graph is shifted up by \(c\) units.
• When the function is \(y - c = f(x)\), it is shifted down by \(c\) units.

With the vertical shift applied to \(-\sqrt{x}\), the new graph \(y = 1 - \sqrt{x}\) still contains the reflection but now is elevated on the y-axis, altering where it crosses the y-axis, but not its overall symmetry.

Square Root Function

The square root function is an important basic function defined as \(y = \sqrt{x}\). It describes a unique curve, which starts at the origin (0,0) and gradually increases as \(x\) gets larger.
This function is a typical example of a non-linear function, characterized by its distinctive sloping shape which increases quickly at first and then more slowly.

Like all square root functions, \(y = \sqrt{x}\) is only defined for values \(x \geq 0\) because the square root of negative numbers isn't real in basic algebra. This characteristic gives the graph its half-side appearance.

• The function's domain is \([0, \infty)\), meaning it starts from zero and extends infinitely to the right.
• Its range also starts at zero, \([0, \infty)\), because the square root of any non-negative number is non-negative.

Understanding the basic square root function is crucial for applying transformations, as it serves as the foundation to which reflections and shifts are added to modify the graph's appearance and position on the coordinate plane.