Problem 28 - Thomas Calculus

Question

Exercises \(19-28\) tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ y=1 / x^{2} \quad \text { Left } 2, \text { down } 1 $$

Step-by-Step Solution

Answer

Shift the graph left 2 units to get \(y = \frac{1}{(x+2)^2}\), then shift down 1 unit for the final equation \(y = \frac{1}{(x+2)^2} - 1\).

Step 1: Understand the Transformations

The instruction tells us to shift the graph to the left by 2 units and down by 1 unit. In general, a shift to the left by a units corresponds to replacing \(x\) with \(x+a\) in the function. Also, shifting down by b units corresponds to subtracting b from the function.

Step 2: Apply Horizontal Shift

To shift the graph 2 units to the left, we replace \(x\) with \(x+2\) in the equation. This changes the equation from \(y = \frac{1}{x^2}\) to \(y = \frac{1}{(x+2)^2}\).

Step 3: Apply Vertical Shift

Next, we apply the vertical shift by moving the graph down by 1 unit. This involves subtracting 1 from the current equation, changing it from \(y = \frac{1}{(x+2)^2}\) to \(y = \frac{1}{(x+2)^2} - 1\).

Step 4: Write the Equation for the Shifted Graph

The final equation for the shifted graph, after applying both the horizontal and vertical shifts, is \(y = \frac{1}{(x+2)^2} - 1\).

Step 5: Sketch the Graphs

To sketch the graphs, draw the original graph \(y = \frac{1}{x^2}\), which is a hyperbola with a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y=0\). Then draw the transformed graph \(y = \frac{1}{(x+2)^2} - 1\), which should have the vertical asymptote at \(x=-2\) and the horizontal asymptote at \(y=-1\).

Key Concepts

Horizontal Shift

To understand horizontal shifts in graph transformations, think of moving the entire graph sideways along the x-axis.
This type of transformation can adjust the position of a graph without altering its shape. Here's how it works:

• In essence, if you want to move a function left or right, you modify the x-variable directly in the equation.
• For a shift to the left by "a" units, replace every "x" in the equation with "x+a".
• For example, given the equation \(y = \frac{1}{x^2}\), if we want to shift it 2 units left, we replace "x" with "x+2" which becomes \(y = \frac{1}{(x+2)^2}\).

By applying this transformation, the graph retains its shape but repositions to the horizontal direction desired.
Horizontal shifts are useful in aligning graphs as per new reference points or visual objectives in interpretations.

Vertical Shift

Vertical shifts in graph transformations involve moving the graph up or down along the y-axis. This adjustment alters the graph's position vertically, still preserving its original geometric shape.
Here's how vertical shifts operate:

• Vertical shifts do not modify the x-variable. Instead, they add or subtract directly from the entire function.
• To shift a graph up by \( b \) units, you'd add \( b \) to the function. Conversely, subtracting \( b \) will move it down by that amount.
• For the equation \(y = \frac{1}{(x+2)^2}\), if we need to shift it down by 1 unit, we'd subtract 1 giving us \(y = \frac{1}{(x+2)^2} - 1\).

Vertical shifts are incredibly helpful when trying to adjust graphs to cross specific y-values or ensuring they pass through a particular point on the y-axis.
These shifts are perfect for graphs that need vertical translations to meet certain analytical goals.

Sketching Graphs

Sketching graphs is a skill that combines various transformations to visually represent mathematical functions.
The primary objectives when sketching are accuracy and clarity—in showing both the original and transformed graphs. Here's a structured approach:

• Start by sketching the original function graph. In this case, \(y = \frac{1}{x^2}\) is a hyperbola with its characteristic vertical asymptote at \(x=0\) and horizontal asymptote at \(y=0\).
• Apply the transformations sequentially. First, graph the horizontal shift by moving everything consistently; the initial hyperbola shifts left to have its vertical asymptote now at \(x=-2\).
• Next, incorporate the vertical shift, moving the entire graph down so the horizontal asymptote is \(y=-1\).
• Draw both graphs on the same set of axes and label them respectively for clarity.
• The original equation is represented by its primary hyperbola, and the transformed by its repositioned and downward-shifted counterpart.

Sketching helps convert complex algebraic transformations into visual insights, making them more digestible and comprehensive.
Being able to sketch effectively aids in the greater understanding of how mathematical functions behave when transformed.