Problem 6 - College Algebra with Corequisite Support

Question

The graph of \(f(x)=(1.68)^{x}\) is shifted right 3 units, stretched vertically by a factor of 2, reflected 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

The function is \(g(x) = -2(1.68)^{x-3} - 3\), with y-intercept \(-3.419\), domain \((-∞, ∞)\), and range \((-∞, -3)\).

Step 1: Horizontal Shift

The function \(f(x) = (1.68)^{x}\) is shifted right by 3 units. To apply this horizontal shift, we replace \(x\) with \(x - 3\), resulting in a new function: \( f(x - 3) = (1.68)^{x-3} \).

Step 2: Vertical Stretch

Next, the function is stretched vertically by a factor of 2. To achieve this, we multiply the entire function by 2: \( 2 \cdot (1.68)^{x-3} \).

Step 3: Reflection about the x-axis

The function is then reflected about the \(x\)-axis. To reflect it, we multiply the entire expression by -1: \( -2 \cdot (1.68)^{x-3} \).

Step 4: Vertical Shift

Finally, the function is shifted downward by 3 units by subtracting 3 from the entire expression: \( g(x) = -2 \cdot (1.68)^{x-3} - 3 \).

Step 5: Calculate y-intercept

To find the \(y\)-intercept, we set \(x = 0\) in the equation of \(g(x)\) and solve for \(g(0)\): \[ g(0) = -2 \cdot (1.68)^{-3} - 3 \] First calculate \((1.68)^{-3}\): \[ (1.68)^{-3} = 0.2097 \] (approximately). Then compute \(g(0)\): \[ g(0) = -2 \times 0.2097 - 3 \approx -0.4194 - 3 \approx -3.419 \] (rounded to the nearest thousandth).

Step 6: Determine Domain and Range

The domain of the function \(g(x)\) is the set of all real numbers, \((- ext{infinity}, ext{infinity})\), since exponential functions are defined everywhere on the real line. The range is determined by the transformations: after the reflection, vertical shift, and downward translation, the range of \(g(x)\) becomes \((- ext{infinity}, -3)\), since the exponential function approaches \(-3\) but never reaches it.

Key Concepts

Horizontal Shift

When working with transformations of functions, a horizontal shift changes the position of the graph along the x-axis.
To shift the function to the right, we replace every occurrence of \(x\) with \(x - a\), where \(a\) is the number of units shifted. In our exercise, \(x\) is replaced with \(x - 3\), moving the graph of \(f(x) = (1.68)^x\) three units to the right. This results in the new function being \(f(x - 3) = (1.68)^{x-3}\).

• Example: For \(g(x) = (x-3)^2\), a shift of three units to the right changes the turning point from \((0, 0)\) to \((3, 0)\).
• Visualize: Imagine sliding the entire graph three steps to the right without changing its shape.

Understanding horizontal shifts helps in correctly positioning graphs when writing new function formulas.

Vertical Stretch

A vertical stretch alters the height of a graph, making it 'taller' or 'shorter.'
In this transformation, every y-value of the function is multiplied by the stretch factor. For our function, the stretch factor is 2, which means we multiply the entire function \((1.68)^{x-3}\) by 2.
This results in the new formula being \(2 \cdot (1.68)^{x-3}\).

• A factor greater than 1, like 2, stretches the graph vertically, making it appear taller.
• Conversely, a factor between 0 and 1 compresses the graph vertically.

The effect of a vertical stretch is that distances from the x-axis increase, enhancing the graph's features vertically.

Reflection

Reflection in mathematics involves flipping a graph across an axis.
When the function is reflected about the x-axis, every y-value becomes its opposite. This is achieved by multiplying the whole function by -1. For our function, this transformation changes the expression to \(-2 \cdot (1.68)^{x-3}\).

• Visually, each point on the graph is mirrored vertically across the x-axis.
• Reflection changes the direction in which the graph's curve opens. In this case, upwards to downwards.

This operation is significant in altering the direction and extremity of a function's values.

Domain and Range

The domain and range are crucial aspects of understanding any function.
The domain refers to all the input values \(x\) for which the function is defined. Here, exponential functions like \(g(x) = -2 \cdot (1.68)^{x-3} - 3\) are defined for all real numbers, so the domain is \((-\infty, \infty)\).

The range defines all the possible output values \(g(x)\).
Due to the reflection and downward shift, the graph approaches the line \(y = -3\) but never reaches it. Hence, the range of our function is \((-\infty, -3)\).

• The original exponential function \(f(x) = (1.68)^{x}\) has a range of \((0, \infty)\).
• Transformations, such as reflections and vertical shifts, influence these values significantly.

Grasping domain and range helps us understand the extent of a function's behavior on a graph.