Problem 65 - College Algebra

Question

Write a formula for the function g that results when the graph of a given toolkit function is transformed as described. The graph of \(f(x)=\frac{1}{x^{2}}\) is vertically compressed by a factor of \(\frac{1}{3},\) then shifted to the left 2 units and down 3 units.

Step-by-Step Solution

Answer

The transformed function is \( g(x) = \frac{1}{3(x+2)^2} - 3 \).

Step 1: Identify the Base Function

The base function given is \[ f(x) = \frac{1}{x^2} \]This function represents the graph of a reciprocal function squared, which is characterized by a hyperbola shape centered around the origin.

Step 2: Apply Vertical Compression

To vertically compress the function by a factor of \( \frac{1}{3} \), multiply the entire function by \( \frac{1}{3} \).Thus, the new function becomes: \[ f_1(x) = \frac{1}{3} \cdot \frac{1}{x^2} = \frac{1}{3x^2} \]

Step 3: Shift Left by 2 Units

To shift the function left by 2 units, replace \( x \) with \( x + 2 \) in the function.This gives us: \[ f_2(x) = \frac{1}{3(x+2)^2} \] This shift means the center of the hyperbola moves two units to the left on the x-axis.

Step 4: Shift Down by 3 Units

To shift the function down by 3 units, subtract 3 from the entire function.The final equation becomes: \[ g(x) = \frac{1}{3(x+2)^2} - 3 \]This shift corresponds to moving the graph downward by 3 units, affecting the y-intercepts.

Key Concepts

Vertical Compression

A vertical compression is a transformation that "squashes" the graph of a function towards the x-axis. This is different from a vertical stretch, which "pulls" the graph away from the x-axis.
For example, if we take the function \( f(x) = \frac{1}{x^2} \), a vertical compression by a factor of \( \frac{1}{3} \) means each point on the graph will be one-third of its original height above the x-axis.
This results in a "flatter" graph.

• To apply vertical compression, you multiply the entire function by the factor \( \frac{1}{3} \).
• The new function becomes \( f_1(x) = \frac{1}{3x^2} \). This affects the steepness of the curve.

Understand that this only affects the y-values, leaving the x-values unchanged.

Horizontal Shift

A horizontal shift involves moving the entire graph of a function left or right along the x-axis. It's like sliding the graph sideways.
In transforming \( f_1(x) = \frac{1}{3x^2} \), shifting it 2 units left requires a replacement of \( x \) in the function with \( x+2 \).

• This transformation results in the function \( f_2(x) = \frac{1}{3(x+2)^2} \).
• Each point moves 2 units to the left.

This does not change the graph's shape, only its position relative to the origin.

Vertical Shift

A vertical shift moves the graph up or down along the y-axis. It's often a straightforward operation of adding or subtracting a constant from the function.
For the graph \( f_2(x) = \frac{1}{3(x+2)^2} \), shifting it down 3 units involves subtracting 3.

• The updated function is now \( g(x) = \frac{1}{3(x+2)^2} - 3 \).
• This means every point on the graph drops by 3 units, affecting the y-intercepts and the range of the function.

This maintains the shape and horizontal placement, solely affecting the vertical location.

Reciprocal Function

A reciprocal function has the general form \( f(x) = \frac{1}{x} \) or forms similar like \( f(x) = \frac{1}{x^2} \). These functions are known for their distinctive shape, usually a hyperbola.
The reciprocal function \( f(x) = \frac{1}{x^2} \) tends towards the x-axis as \( x \) increases or decreases.

• They do not cross the x- or y-axes, which means they have asymptotes at these lines.
• The transformations explained above manipulate this unique shape either by compressing it, or shifting it along the axes.

These characteristics make understanding transformations crucial; they can dictate how the graph fits within the Cartesian coordinate system.