Problem 34 - Thomas Calculus

Question

Exercises \(25-34\) tell how many units and in what directions the graphs of the given equations are 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

The graph is shifted left by 2 units and down by 1 unit. The new equation is \( y = \frac{1}{(x+2)^2} - 1 \).

Step 1: Identify the translation directions

The function is to be shifted left by 2 units and down by 1 unit. Translating a function to the left by 2 units requires replacing \(x\) with \(x+2\) in the function. Translating the function down by 1 unit requires subtracting 1 from the function.

Step 2: Modify the equation for horizontal shift

The original equation is \( y = \frac{1}{x^2} \). To shift this graph to the left by 2 units, we replace \( x \) with \( x + 2 \). The modified equation becomes \( y = \frac{1}{(x+2)^2} \).

Step 3: Modify the equation for vertical shift

After applying the horizontal shift, we need to shift the graph down by 1 unit. This is done by subtracting 1 from the current equation. The updated equation becomes \( y = \frac{1}{(x+2)^2} - 1 \).

Step 4: Sketch the graphs and label them

In the sketch, draw the original graph of \( y = \frac{1}{x^2} \), which is a simple curve that approaches the x-axis asymptotically from both sides of the y-axis. Draw the shifted graph \( y = \frac{1}{(x+2)^2} - 1 \), which is this curve moved 2 units to the left and 1 unit downward. Make sure to label each graph with its corresponding equation.

Key Concepts

Translation

Graph translation is a transformation that shifts the graph to a different position in the coordinate plane. When translating a graph, its shape and orientation remain unchanged. We only move it either vertically, horizontally, or both. To achieve this, changes are made to the equation of the graph.

• A horizontal translation moves the graph left or right.
• A vertical translation moves the graph up or down.

By replacing variables in the function's equation, we can control how the graph is shifted in the coordinate plane. For example, replacing \( x \) with \( x + a \) shifts the graph horizontally. Similarly, adding or subtracting a constant from the full function translates it vertically.

Vertical shift

A vertical shift involves moving a graph up or down without altering its shape. To implement a vertical shift, we adjust the entire function up or down by a specific number of units. In mathematical terms, this is done by adding or subtracting a constant from the whole function.
If you want to shift the graph of a function \( f(x) \) upwards by \( k \) units, you modify the equation to \( f(x) + k \). Similarly, to move it downward by \( k \) units, you change it to \( f(x) - k \).
Example:

• Given \( y = \frac{1}{x^2} \)
• Shifting it down by 1 unit results in \( y = \frac{1}{x^2} - 1 \)

This changes its vertical position without affecting how the curve looks or its direction.

Horizontal shift

Horizontal shifts move a graph left or right along the x-axis. By modifying the x-variable in the function's equation, we dictate the direction and magnitude of this shift. This type of transformation is crucial when we want to reposition a graph without altering its shape or vertical placement.
To shift a function left by \( h \) units, you replace \( x \) with \( x + h \) in the equation. On the other hand, for a rightward shift of \( h \) units, replace \( x \) with \( x - h \).
For instance, starting with \( y = \frac{1}{x^2} \):

• Shifting it left by 2 units changes it to \( y = \frac{1}{(x+2)^2} \)

This modification keeps the curve's overall shape and slope constant, simply relocating it horizontally in the plane.

Equation modification

Modifying the equation of a function is the heart of graph transformations, allowing us to shift, stretch, or compress graphs as needed. Each type of transformation requires specific changes to the function's equation parameters, focusing on either the x-variable or the entire function.
For translations:

• Horizontal translations involve adjusting the x-variable: \( x \rightarrow x + h \) for left shift, \( x \rightarrow x - h \) for right shift.
• Vertical translations adjust the entire equation: adding a constant for an upward shift and subtracting for downward movement.

These transformations do not change the graph's shape. They only relocate it on the graph plane. By correctly modifying the equations, we could easily control graph movements in both the x and y directions, tailoring the graph’s position to fit our needs.