Problem 5 - College Algebra

Question

The graph of \(f(x)=10^{x}\) is refl cted about the \(x\) -axis and shifted upward 7 units. 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 \( g(x) = -10^x + 7 \). Y-intercept: 6. Domain: all real numbers. Range: \((-\infty, 7)\).

Step 1: Reflect the Function Across the X-Axis

Reflecting a function across the x-axis means multiplying the function by -1. The original function is \( f(x) = 10^x \). After reflection, the function becomes \( g(x) = -10^x \).

Step 2: Shift the Function Upward

To shift the function 7 units upward, we add 7 to the reflected function. So, the function becomes \( g(x) = -10^x + 7 \).

Step 3: Determine the Y-Intercept

The y-intercept is the point where the function crosses the y-axis, i.e., when \( x = 0 \). Substitute \( x = 0 \) into \( g(x) = -10^x + 7 \), which gives \( g(0) = -10^0 + 7 = -1 + 7 = 6 \). Thus, the y-intercept is \( 6 \).

Step 4: Determine the Domain

The domain of \( g(x) = -10^x + 7 \) is the set of all real numbers, as there are no restrictions on the values that \( x \) can take.

Step 5: Determine the Range

The range of the function \( g(x) = -10^x + 7 \) starts just below 7 and decreases to negative infinity. As \( x \) becomes large, \( 10^x \) approaches infinity, so \( -10^x \) approaches negative infinity, thus the function does as well. The range is \((-\infty, 7)\).

Key Concepts

Reflection Across the X-Axis

In mathematics, reflecting a function across the x-axis involves taking the graph of the function and flipping it over the x-axis. This is done by multiplying the entire function by

• -1.

Think of it as creating a mirror image of the curve below the x-axis. For example, if you start with the function \(f(x) = 10^x\), a reflection across the x-axis transforms it into \(g(x) = -10^x\).

This reflection process affects how the graph looks because every point that was originally above the x-axis is now below it. Thus, positive output values become negative, leaving the shape of the graph unchanged except for its orientation around the x-axis.

Vertical Shift

A vertical shift in function transformation involves moving the entire graph of a function up or down along the y-axis.

This shift does not alter the shape of the graph, only its position.

• To shift a function upward, add the desired number of units to the function.
• If shifting downward, subtract the units.

For instance, after reflecting the original function \(-10^x\) from the previous section, shifting upwards by 7 units means adding 7 to this transformed function. This results in the new equation \(g(x) = -10^x + 7\).

This transformation raises every point on the graph by 7 units. More intuitively, you can envision each point on the graph moving straight up but maintaining the same x coordinate.

Exponential Functions

Exponential functions, such as \(f(x) = 10^x\), are characterized by a constant base raised to a variable exponent.

• These functions grow extremely quickly, as the value of x increases the function's output increases exponentially.
• In the context of transformations, exponential functions are particularly interesting because of how rapidly they evolve.

When applying transformations like reflections and shifts, you modify how steeply the function rises or falls. The reflection yields \(-10^x\) which decreases rapidly as x increases, whereas the vertical shift adjusts its position but not the rate of its decrease. The fascinating aspect of working with transformed exponential functions like \(-10^x + 7\) is understanding their behavior under these transformations. Even slight changes in these functions can result in large alterations in their appearance and range. For this specific function, its domain remains the set of all real numbers, and its range is transformed to \((-\infty, 7)\), demonstrating the typical behavior of transformed exponential functions.