Problem 6 - College Algebra

Question

The graph of \(f(x)=(1.68)^{x}\) is shifted right 3 units, stretched vertically by a factor of \(2,\) refl cted about the \(x\) -axis, and then shifted downward 3 units. What is the equation of the new function, \(g(x) ?\) State its \(y\) -intercept (to the nearest thousandth), domain, and range.

Step-by-Step Solution

Answer

\( g(x) = -2(1.68)^{x-3} - 3 \), \( y \)-intercept: \(-3.420\), domain: all real numbers, range: \((-\infty, -3)\).

Step 1: Horizontal Shift

To shift the graph of the function to the right by 3 units, replace every instance of \( x \) in the function \( f(x) = (1.68)^x \) with \( x - 3 \). This gives us \( f(x-3) = (1.68)^{x-3} \).

Step 2: Vertical Stretch

To stretch the graph vertically by a factor of 2, multiply the entire function by 2. Thus, \( 2(1.68)^{x-3} \) is the transformed function after a vertical stretch.

Step 3: Reflection Across the x-Axis

Reflect the function across the \( x \)-axis by multiplying it by -1. This transforms the function into \( -2(1.68)^{x-3} \).

Step 4: Vertical Shift

Shift the function downward by 3 units by subtracting 3 from the entire function. The equation becomes \( g(x) = -2(1.68)^{x-3} - 3 \).

Step 5: Determine the y-Intercept

To find the \( y \)-intercept, evaluate \( g(x) \) at \( x = 0 \): \( g(0) = -2(1.68)^{0-3} - 3 = -2(1.68)^{-3} - 3 \). Calculate \( (1.68)^{-3} \) and simplify to find the \( y \)-intercept.

Step 6: Calculate y-Intercept and Simplify

\( (1.68)^{-3} = \frac{1}{(1.68)^3} \approx 0.210\). So, \( g(0) = -2 \times 0.210 - 3 = -0.42 - 3 = -3.42 \). Thus, the \( y \)-intercept is approximately \(-3.420\).

Step 7: Determine Domain and Range

The domain of \( g(x) = -2(1.68)^{x-3} - 3 \) is all real numbers because there are no restrictions on \( x \) for exponential functions. The range is \((-\infty, -3)\) because the function is reflected and shifted downward, causing it to never reach or exceed \(-3\).

Key Concepts

Function Transformation

Function transformations are alterations applied to basic functions to change their shape, position, or orientation. In the context of the exercise, we're transforming the function \( f(x) = (1.68)^x \). Understanding these transformations helps in visualizing graph changes easily.

There are several types of transformations:

• Horizontal Shifts: moving the graph left or right
• Vertical Shifts: moving the graph up or down
• Reflections: flipping the graph over an axis
• Stretches and Shrinks: changing the size of the graph

For each transformation, specific alterations are made to the function's equation. For example, shifting a function horizontally involves changing the input variable \( x \). Meanwhile, vertical stretches affect how steep the graph appears.

Domain and Range

The domain and range of a function provide important information about what values are acceptable as inputs and what outputs those inputs produce. For exponential functions like \( f(x) = (1.68)^x \), the domain includes all real numbers. In simpler terms, you can plug any number into the function.

The range, however, might change depending on transformations. In the original function \( (1.68)^x \), outputs are always positive. But after applying the transformations given in the problem, the range changes. Due to the reflection over the x-axis and a downward shift, the range becomes \((-\infty, -3)\). This tells us that the function never outputs a value of -3 or higher.

Graph Shifts

When performing graph shifts, you're moving the entire graph without altering its shape. A horizontal shift involves moving left or right. In our example, this is done by replacing \( x \) with \( x - 3 \), indicating a rightward shift.

Vertical shifts either raise or lower the graph. To shift down by 3 units, we subtract 3 from the function. This results in the final equation, \( g(x) = -2(1.68)^{x-3} - 3 \). Graph shifts are intuitive: think of moving a physical object along a surface. You're changing the position but not its dimensions or orientation.

Reflection and Stretching

Reflections and stretching are more advanced transformations compared to shifts. They change the graph's shape and perspective, giving it a new orientation or size.

Reflecting a function over the \( x \)-axis flips it upside down. This is done by multiplying the entire function by -1, transforming the outputs from positive to negative values.

Stretching involves enlarging the graph in a particular direction. A vertical stretch by a factor of 2, for instance, multiplies every output by 2. This makes the peaks higher and the troughs lower along the y-axis. In the exercise, this gives us a steeper graph before further transformations are applied. These techniques allow for intricate modifications that can tailor a function to specific needs and applications.