Problem 32 - 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. $$f(x)=\frac{-1}{(x-4)^{2}}+2$$

Step-by-Step Solution

Answer

Shift right 4, reflect x-axis, shift up 2; Domain: \(x \neq 4\); Range: \((-\infty, 2)\).

Step 1: Identify Base Graph

The given function is a transformation of the base graph \(y = \frac{1}{x^2}\). This is an important note, as it will help guide the transformations we apply.

Step 2: Apply Horizontal Shift

The function \(f(x) = \frac{-1}{(x-4)^{2}} + 2\) includes a horizontal shift component: \(x - 4\). This means that the graph of \(y = \frac{1}{x^2}\) is shifted 4 units to the right.

Step 3: Apply Vertical Stretch/Shrink and Reflection

The coefficient of \(-1\) in \(\frac{-1}{(x-4)^2}\) indicates two transformations: the graph reflects across the x-axis and there is a vertical stretch. This changes the graph, flipping it upside down.

Step 4: Apply Vertical Shift

The constant \(+2\) at the end of the function indicates the vertical shift upwards by 2 units. This moves the entire graph up by 2 units along the y-axis.

Step 5: Sketch the Graph

Combine all transformations. Start from \(y = \frac{1}{x^2}\), shift right by 4, reflect across the x-axis, and move up by 2 units. Sketch this on graph paper.

Step 6: Domain and Range

The domain of \(f(x) = \frac{-1}{(x-4)^{2}} + 2\) is all real numbers except \(x = 4\), because \((x-4)^2\) cannot be zero. The range of this function is \((-\infty, 2)\), because the graph has a horizontal asymptote at \(y = 2\), and the graph never actually reaches 2.

Key Concepts

Horizontal Shift

When examining the function \(f(x) = \frac{-1}{(x-4)^{2}} + 2\), one of the key transformations is the horizontal shift. A horizontal shift moves the graph left or right along the x-axis.

• The component \(x-4\) in the denominator denotes this horizontal shift.
• Specifically, it tells us that the graph of the base function \(y = \frac{1}{x^2}\) is shifted 4 units to the right.

Typically, an expression within a function \( (x-h) \) moves the graph to the right if \(h\) is positive and to the left if \(h\) is negative. In this case, the \(-4\) indicates a right shift, moving every point on the graph 4 units over along the x-axis. This transformation ensures the symmetrical properties of the parent graph \(y = \frac{1}{x^2}\) remain intact but are repositioned horizontally.

Vertical Stretch

Another transformation present in \(f(x) = \frac{-1}{(x-4)^{2}} + 2\) is the vertical stretch. A vertical stretch occurs when all the y-coordinates of a graph are multiplied by a factor, making the graph appear taller or shrunk.

• The multiplier \(-1\) outside the fraction performs this stretch, but it also affects graph directionality.
• Though the stretch is present, the primary outcome of \(-1\) is reflecting the graph over the x-axis, which we will dig into shortly.

Sometimes, numbers greater than 1 cause vertical stretching, while numbers between 0 and 1 result in vertical shrinking. Despite the presence of the \(-1\) causing the graph to flip, the function \(y = \frac{1}{x^2}\) still maintains its essential characteristics, but with an altered visual footprint in the vertical axis.

Reflection

In the equation \(f(x) = \frac{-1}{(x-4)^{2}} + 2\), the reflection component arises from the \(-1\) in front of the fraction. This coefficient executes a reflection across the x-axis.

• Reflection transforms the graph upside down.
• Every y-coordinate value of the graph \(y = \frac{1}{x^2}\) is converted from positive to negative, effectively flipping it below the x-axis.

This fundamental change affects the narrative of the graph. While it still exhibits the familiar bell-like shape characteristic of \(\frac{1}{x^2}\), instead of opening upwards, the graph opens downwards. This reflection gives the graph a different dynamic across the x-axis while maintaining the same horizontal and vertical dimensions.

Vertical Shift

The final key transformation in the function \(f(x) = \frac{-1}{(x-4)^{2}} + 2\) involves a vertical shift. A vertical shift moves the entire graph up or down along the y-axis.

• The constant \(+2\) at the end of the function suggests a shift 2 units upwards.
• It lifts every point on the graph up by 2 units, repositioning the asymptotic behavior to reflect above the x-axis.

Vertical shifts preserve the symmetry and shape of the original graph, just lifting or sinking the whole setup. In our function, the domain remains unchanged, except its horizontal asymptote, due to the shift, is now at \(y = 2\) instead of the traditional \(y = 0\). Understanding each transformation step helps visualize the complete picture of how \(f(x) = \frac{-1}{(x-4)^{2}} + 2\) evolves from \(y = \frac{1}{x^2}\).