Problem 496 - Precalculus

Question

The graph of \(f(x)=6.5^{x}\) is reflected about the \(y\) -axis and stretched vertically by a factor of \(7 .\) What is the equation of the new function, \(g(x) ?\) State its \(y\) -intercept, domain, and range.

Step-by-Step Solution

Answer

The equation is \(g(x) = 7 \, 6.5^{-x}\); y-intercept: 7; domain: all real numbers; range: (0, ∞).

Step 1: Reflect around the y-axis

To reflect a function across the y-axis, we replace \(x\) with \(-x\) in the original function equation. Thus, reflecting \(f(x) = 6.5^x\) across the y-axis results in \(f(-x) = 6.5^{-x}\).

Step 2: Vertical Stretch by a Factor of 7

To stretch a function vertically by a factor of 7, we multiply the entire function by 7. So, applying this to \(f(-x) = 6.5^{-x}\), the new function becomes \(g(x) = 7 \, \cdot \, 6.5^{-x}\).

Step 3: Identify the y-intercept

The y-intercept occurs when \(x = 0\). Plugging \(x = 0\) into \(g(x) = 7 \, \cdot \, 6.5^{-x}\), we get \(g(0) = 7 \, \cdot \, 6.5^{0} = 7 \, \cdot \, 1 = 7\). So, the y-intercept of \(g(x)\) is 7.

Step 4: Determine the Domain

The domain of \(g(x) = 7 \, \cdot \, 6.5^{-x}\) is all real numbers, \(x \in (-\infty, \infty)\), because there are no restrictions on the exponential function in terms of \(x\).

Step 5: Determine the Range

The range of \(g(x) = 7 \, \cdot \, 6.5^{-x}\) corresponds to the set of possible output values. Since exponential functions of the form \(b^{-x}\) (for \(b>1\)) decay from 1 to 0 as \(x\) increases, the range of \(7 \, \cdot \, 6.5^{-x}\) is \((0, \infty)\), reflecting both the exponential decay and the vertical stretch.

Key Concepts

Transformation of Functions

Understanding the transformation of functions is crucial when manipulating graphs of equations. When you transform a function, you change its position, size, orientation, or shape in a graph. For the function given in the exercise, the transformation includes a reflection and a vertical stretch.

Reflecting and stretching are common types of transformations used in graphs.

• Translations: Shift the graph horizontally or vertically without changing its shape.
• Reflections: Flip the graph over a specific line.
• Stretches and Compressions: Change the size of the graph either vertically or horizontally.

In our case, the transformation involves specifically reflecting the function across the y-axis and stretching it vertically by a given factor. Each type of transformation affects the graph in a predictable way, altering its mathematical equation accordingly.

Vertical Stretch

A vertical stretch involves elongating the graph of a function along the vertical axis. This type of transformation modifies the "height" of the function without affecting its "width". In the exercise, the function was stretched vertically by a factor of 7.

To perform a vertical stretch:- Multiply the function by the stretch factor.- This multiplication applies to the entire function.For example, if you have a function \(f(x) = 6.5^x\), stretching it vertically by 7 means you multiply the entire function, giving you \(g(x) = 7 \cdot 6.5^{-x}\).
Note that this transformation does not change the x-values or move the graph up or down—it only makes it taller or shorter, depending on the factor used.

Reflection Across Y-Axis

A reflection across the y-axis means flipping the graph over this vertical line. In mathematical terms, to reflect a function over the y-axis, you replace every instance of \(x\) with \(-x\) in the function’s equation.

For the function \(f(x) = 6.5^x\), replacing \(x\) with \(-x\) results in \(f(-x) = 6.5^{-x}\). This operation effectively flips the graph from side to side around the y-axis.
The reflection changes the direction of growth or decay in exponential functions:

• Instead of growing rapidly as \(x\) increases, a reflected exponential function like \(6.5^{-x}\) approaches zero.
• This transformation also inversely affects the domain direction: even though the domain remains all real numbers, the function behaves differently as x becomes more positive or negative.

This concept is particularly important for understanding how the orientation of a graph can be flipped while maintaining its shape and the relationships between its points.