Problem 40 - College Algebra with Modeling and Visualization

Question

Use transformations to explain how the graph of \(f\) can be found by using the graph of \(y=x^{2}\) \(y=\sqrt{x},\) or \(y=|x| .\) You do not need to graph \(y=f(x)\). \(f(x)=-\sqrt{x}-3\)

Step-by-Step Solution

Answer

Start with \(y = \sqrt{x}\), reflect over the x-axis, then shift downward by 3 units.

Step 1: Identify the Base Function

The base function of the given function is identified. Here, the base function is \(y = \sqrt{x}\). This is the function that undergoes transformations to become \(f(x)\).

Step 2: Apply Reflection

Identify that the coefficient of the square root is negative, \(- \sqrt{x}\). This implies a reflection of the graph over the \(x\)-axis, flipping it upside down.

Step 3: Apply Vertical Shift

Recognize the \(-3\) outside of the square root. This means that the graph shifts vertically downward by 3 units. Thus, every point on the graph after reflection is moved 3 units down.

Key Concepts

Base Function

A **base function** is the starting point when considering transformations. It's like having a blank canvas to start our paintwork. In our case, the base function is given as \( y = \sqrt{x} \). This function describes a curve that starts at the origin and rises gently to the right. It’s important to familiarize yourself with the graphs of common base functions, so you have a clear picture in your mind of how they look.

In transformations, once you've identified the base function, the next steps involve changing this graph to fit the new function. The process involves transformations such as stretches, shifts, and reflections.

As you tackle these transformations, always start by identifying the base function. It sets the stage for all the adjustments you are going to make to the graph.

Reflection

**Reflection** is a kind of transformation that flips the graph of a function across a line, like a mirror image. For the function \( f(x) = -\sqrt{x} - 3 \), the negative sign in front of \( \sqrt{x} \) indicates a reflection over the \( x \)-axis.

This means turning the curve upside down. Imagine flipping a spoon on a table—the curve this time bends downward. Graphically, every point on the original graph \( y = \sqrt{x} \) is transformed to its opposite height, so values above the \( x \)-axis are mapped below it and vice versa.

Understanding this step is crucial because it significantly alters how the graph looks. If you see a negative sign before the function, expect a reflection. It's a common trick question point in exams, so always watch out for it!

Vertical Shift

A **vertical shift** moves a graph up or down without changing its shape. When you have the function \( f(x) = -\sqrt{x} - 3 \), the "\(-3\)" is crucial. Here, it indicates a shift downward by 3 units.

Think of this as adjusting a painting on a wall to hang a little lower. It doesn't change the painting itself, just where it hangs in your field of vision. In practical terms, every point on the reflected curve \( -\sqrt{x} \) is moved 3 units down.

Vertical shifts are straightforward yet vital for positioning graphs correctly. This shift is crucial as it finalizes the location of the graph. Always be alert for constant terms outside the main function expression as they often control these shifts.