Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}} .\) Draw a sketch of the graph of \(f\) by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range. $$f(x)=\frac{-1}{(x+2)^{2}}-3$$

Step-by-Step Solution

Answer

The graph is an upside-down \(\frac{1}{x^2}\) shape shifted left by 2 units and down by 3 units. Domain: \((-\infty, -2) \cup (-2, \infty)\), Range: \((-\infty, -3)\).

Step 1: Identify the Base Graph

The function you've been given is transformed from the base function \(y = \frac{1}{x^2}\). This base graph has a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 0\).

Step 2: Horizontal Shift

The expression \((x+2)\) in the denominator indicates a horizontal shift. The graph of \(y = \frac{1}{x^2}\) moves two units to the left, resulting in the new function \(y = \frac{1}{(x+2)^2}\). The new vertical asymptote is at \(x = -2\).

Step 3: Vertical Stretch and Reflection

The negative sign in front of the fraction, \(-\frac{1}{(x+2)^2}\), reflects the graph across the x-axis. As a result, it is flipped upside-down compared to the standard \(y = \frac{1}{x^2}\) graph, turning the entire graph upside down.

Step 4: Vertical Shift

The term \(-3\) at the end of the function translates the graph downward by three units. This affects the horizontal asymptote, moving it from \(y = 0\) to \(y = -3\).

Step 5: Sketch the Graph

Start by drawing the base graph of \(y = \frac{1}{x^2}\) as a reference. Apply the transformations step by step: move left by two units, reflect it in the x-axis, and then shift it down by three units. Ensure the vertical asymptote is at \(x = -2\) and the horizontal asymptote is at \(y = -3\).

Step 6: Determine Domain and Range

For the function \(f(x) = \frac{-1}{(x+2)^{2}} - 3\), the domain includes all real numbers except where the function is undefined, which is at \(x = -2\). Therefore, the domain is \((-\infty, -2) \cup (-2, \infty)\). The range was affected by the vertical transformations and is \((-\infty, -3)\).

Key Concepts

Domain and Range

The domain of a function refers to all the possible x-values that can be input into the function, resulting in a real number output. For the function \(f(x) = \frac{-1}{(x+2)^{2}} - 3\), the denominator \((x+2)^{2}\) cannot equal zero, as division by zero is undefined. Hence, \(x\) cannot be \(-2\). This gives us the domain:

• All real numbers except \(x = -2\), written as \((-finity, -2) \cup (-2, \infty)\).

The range of a function is determined by the set of possible y-values the function can produce. The negative sign in front of the fraction and the \(-3\) at the end indicate the function values can extend infinitely downward, from negative infinity, to just less than \(-3\). Thus, the range is

• \((-fty, -3)\).

Vertical Asymptotes

Vertical asymptotes occur where the function becomes undefined and the graph shoots up or down toward infinity. For \(f(x) = \frac{-1}{(x+2)^{2}} - 3\), the vertical asymptote emerges where the denominator is zero, specifically at \(x = -2\).

• This means the graph will approach but never touch or cross the line \(x = -2\).

Understanding vertical asymptotes is crucial because they show where the function behaves dramatically, tending toward extreme values.
The asymptote reflects fundamental changes in the function's continuity, shaping the graph's sharp bend and break at \(x = -2\).

Horizontal Shifts

Horizontal shifts in graphs occur when you adjust the x-values in a function. For \(y = \frac{1}{x^2}\), altering it to \(y = \frac{1}{(x+2)^2}\) pushes the graph to the left by two units.

• The graph is essentially being slid along the x-axis without changing its shape.
• This leads to the shifted vertical asymptote, now present at \(x = -2\).

Grasping horizontal shifts helps in understanding how certain elements like asymptotes and intercepts move, keeping the same relative orientation. It’s like moving a whole picture frame horizontally along a wall.

Reflections

Reflections in functions occur when there's a negative sign affecting the entire function or part of it. In \(f(x) = \frac{-1}{(x+2)^{2}} - 3\), the negative coefficient out front

• inverts the graph across the x-axis.

This reflection transforms what would typically be a positive range of \(y\)-values (as seen in \(y = \frac{1}{x^2}\)) into negative ones in our transformed function.
Reflections adjust the graph's orientation, creating an upside-down effect compared to its typical form. Understanding reflections allows one to predict how the function's behavior will change, such as peaks turning into troughs when flipped across a line.