Problem 9 - College Algebra

Question

For the following exercises, write a formula for the function obtained when the graph is shifted as described. \(f(x)=\frac{1}{x^{2}}\) is shifted up 2 units and to the left 4 un ts.

Step-by-Step Solution

Answer

The function is \( f(x) = \frac{1}{(x+4)^2} + 2 \).

Step 1: Understanding the Original Function

The original function given is \( f(x) = \frac{1}{x^2} \). This is a rational function which tends to infinity as \( x \) approaches 0 from either direction and tends to 0 as \( x \) tends towards infinity or negative infinity.

Step 2: Vertical Shift Up 2 Units

A vertical shift up involves adding a constant to the function. For a shift up of 2 units, we add 2 to the function. The function changes from \( f(x) = \frac{1}{x^2} \) to \( f(x) + 2 = \frac{1}{x^2} + 2 \).

Step 3: Horizontal Shift Left 4 Units

A horizontal shift left involves replacing \( x \) with \( x + k \) for a shift left by \( k \) units. Here, \( k = 4 \). Therefore, replace \( x \) with \( x + 4 \) in the function. The function changes to \( f(x) = \frac{1}{(x+4)^2} + 2 \).

Step 4: Writing the Final Function

Combine the transformations. The final formula for the function after shifting the original graph up 2 units and left 4 units is \( f(x) = \frac{1}{(x+4)^2} + 2 \).

Key Concepts

Vertical Shift

A vertical shift involves moving the entire graph of a function up or down without affecting its shape. When we talk about a vertical shift, we are referring to adding or subtracting from the entire function. This transformation shifts the graph on the y-axis.

Here's how it works with the equation given:

• Original Function: \( f(x) = \frac{1}{x^2} \)
• Vertical Shift: Adding 2 to the function moves it up by 2 units.

After the shift, the new function becomes \( f(x) + 2 = \frac{1}{x^2} + 2 \). This means every point of the original function moves 2 units higher on the graph. Graphically, the distances above the x-axis increase by 2, which affects the range. The overall behavior as \( x \) approaches certain values remains unchanged.A vertical shift does not change the domain of the function, but it does alter the range. For the original function \( \frac{1}{x^2} \), the range would have been all positive real numbers. With the shift, the range starts from 2 and goes upward.

Horizontal Shift

A horizontal shift modifies a function by moving it left or right. This type of transformation affects the input value (or x-value) directly.

For a horizontal shift:

• To shift to the left by a specific number of units, we replace \( x \) with \( x + k \) where \( k \) is positive.
• To shift to the right, we replace \( x \) with \( x - k \).

In the given exercise, we shift the function left by 4 units:

• Original Function: \( f(x) = \frac{1}{x^2} \)
• Horizontal Shift: Replace \( x \) with \( x + 4 \).

This results in a new function: \( f(x) = \frac{1}{(x+4)^2} \). This transformation means each point of the original graph shifts to the left by 4 units. The domain of the function can also change or shift accordingly, but crucially, the range will stay the same once adjusted by any vertical shift.

Rational Function

Rational functions are defined as a ratio of two polynomials. A basic form would be \( \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) ot= 0 \).

These types of functions can display a wide range of behaviors based on their polynomials. For example, asymptotes often occur in rational functions. Vertical asymptotes appear where the denominator equals zero, while horizontal or oblique asymptotes can describe end-behavior concerning the highest degree terms of numerator and denominator.In the given function, \( f(x) = \frac{1}{x^2} \):

• The function has a vertical asymptote at \( x = 0 \), meaning as \( x \) approaches 0, \( f(x) \) will approach infinity or negative infinity.
• It also tends to zero as \( x \) approaches either positive or negative infinity.

By applying transformations, we can change the positioning of these asymptotes. A vertical shift will move the horizontal asymptote up or down, and a horizontal shift will move any vertical asymptote left or right. For this specific exercise, moving the graph up 2 units and left 4 units shifts the vertical asymptote to \( x = -4 \) while the effect of the upward shift is that the horizontal asymptote rises to \( y = 2 \). Understanding these shifts allow us to visually track how the graph will change.