Problem 46 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

Reflect the graph of \(y = \sqrt{x}\) across the y-axis and shift it down by 1 unit.

Step 1: Identify the basic function

The basic function involved here is the square root function, which is \(y = \sqrt{x}\). The graph of this function is a curve starting at the origin and increasing slowly to the right along the x-axis.

Step 2: Apply horizontal transformation

Examine the transformation inside the square root, \(-x\). This reflects the graph of \(y = \sqrt{x}\) across the y-axis. This means that instead of the graph being in the first quadrant (where x is positive), it will be in the second quadrant, where x is non-positive.

Step 3: Apply vertical transformation

Next, consider the \(-1\) outside the square root. This shifts the entire graph vertically downwards by 1 unit. Any point \( (a, b) \) on the original \(y = -\sqrt{x} \) will now be \( (a, b-1) \) on the graph of \(y = \sqrt{-x} - 1\).

Step 4: Sketch the transformed graph

Begin by drawing the parent function \(y = \sqrt{x} \) in the first quadrant. Then, reflect it across the y-axis to have it in the second quadrant as \(y = \sqrt{-x} \). Finally, shift this graph down by 1 unit to reflect the transformation to \(y = \sqrt{-x} - 1\). The endpoint of the curve, which was originally at \(0, 0\), is now at \(0, -1\), and the curve extends to the left.

Key Concepts

Square Root Function

The square root function is one of the basic and commonly used functions in mathematics. Its form is given by \(y = \sqrt{x}\). This function produces a curve that originates from the point (0, 0) on the graph, which is called the origin, and progressively rises as it moves to the right along the x-axis. Here are some key characteristics of the square root function:

• It is only defined for x-values that are zero or positive since square roots of negative numbers are not real.
• The graph extends infinitely to the right as x increases, but it does so with a decreasing rate of increase.
• The function is continuous, meaning there are no breaks or jumps in the graph.

Understanding the basic square root function is crucial, as it forms the foundation for applying various transformations, such as reflections or shifts, to sketch internal variations of this curve.

Reflection across y-axis

Reflection across the y-axis is a type of transformation applied to a graph, and it essentially involves flipping the graph over the y-axis. This transformation alters the direction of the graph. To perform this reflection, you apply a negative sign to every instance of \(x\) within the function, turning \(f(x)\) into \(f(-x)\). For example, reflecting the basic square root function \(y = \sqrt{x}\) gives us \(y = \sqrt{-x}\). Here’s what happens after this transformation:

• The graph, which initially resides in the first quadrant (where x is positive), moves to the second quadrant (where x is non-positive).
• The original points on the curve undergo a coordinate swap; for instance, a point at \((a, b)\) becomes \((-a, b)\).
• This reflection drastically changes the visual aspect of the graph, helping in addressing different contexts where negatives of the input values are required.

Reflections are a simple yet powerful tool in graph transformations, enabling the visualization of functions in a different perspective on Cartesian planes.

Vertical Shift

A vertical shift in a graph involves moving all the points of the function up or down along the y-axis by a specific constant. This transformation doesn't alter the shape of the graph; it only changes the position.In the context of \(f(x) = \sqrt{-x} - 1\), the \(-1\) outside the square root signifies a vertical shift. Here’s how you can interpret it:

• The entire graph \(y = \sqrt{-x}\) moves downwards by 1 unit.
• Each y-value from the original function is decreased by 1, shifting the whole curve to new y-positions.
• This results in altering the y-intercept and any vertical alignment on the graph.

Understanding vertical shifts is essential as it directly affects the output range of the function while leaving the x-values unaffected. This transformation is frequently used to adjust graphs to fit different real-world scenarios or to reflect specific characteristics within the dataset.

Graph Sketching

Sketching the graph of a function by hand involves integrating multiple transformations to visualize the final altered path of the function. When constructing the graph of \(y=f(x)\), like \(y = \sqrt{-x} - 1\), it's beneficial to follow a structured approach:1. **Start with the basic function**: Recognize the parent function—here, it's \(y = \sqrt{x}\). Sketch this in the first quadrant.2. **Reflect across the y-axis**: Transform the sketched curve to represent \(y = \sqrt{-x}\). Move it to the second quadrant by flipping it over the y-axis.3. **Apply a vertical shift**: Shift the reflected graph downward by 1 unit to arrive at the desired function, \(y = \sqrt{-x} - 1\).With this structure:

• You begin with a known shape and then apply each transformation step-by-step, making it easier to manage.
• Each transformation affects the graph visually, in a clear and predictable manner.
• The endpoint of the curve shifts due to these transformations, helping to delineate the final, transformed path effectively.

Mastering chart drawing with transformations boosts understanding of graph behaviors and improves analytical skills.