Question

Use transformations to graph each function. Determine the domain, range, horizontal asymptote, and y-intercept of each function. $$ f(x)=1-2^{x+3} $$

Short answer

Domain: \( (-fty, fty) \); Range: \( (-fty, 1) \); Horizontal Asymptote: \( y = 1 \); y-intercept: \( (0, -7) \).

Step-by-step solution

Identify the Parent Function

The parent function is the exponential function of the form \( f(x) = 2^x \).

Translate the Function Horizontally

The term \( x+3 \) indicates a horizontal shift to the left by 3 units. So, the transformed function is \( f(x) = 2^{x+3} \).

Reflect and Translate Vertically

The presence of the negative sign in front of \( 2^{x+3} \) reflects the function across the x-axis, making it \( -2^{x+3} \). Adding 1 translates the function up by 1 unit. Thus, the complete function is \( f(x) = 1 - 2^{x+3} \).

Determine the Horizontal Asymptote

The horizontal asymptote of the transformed function is the horizontal line the graph approaches but never touches. Since we have a vertical translation up by 1, the horizontal asymptote is \( y = 1 \).

Calculate the y-intercept

To find the y-intercept, substitute \( x = 0 \) into the function: \( f(0) = 1 - 2^{0+3} \ = 1 - 2^3 \ = 1 - 8 \ = -7 \). The y-intercept is \( (0, -7) \).

Determine the Domain

The domain of the function \( f(x) = 1 - 2^{x+3} \) is all real numbers \( (-fty, fty) \), since exponential functions are defined for all \( x \).

Determine the Range

The range of the function is the set of all y-values that the function can take. Since the function is vertically reflected and shifted, the range is \( (-fty, 1) \).

Key concepts

Horizontal Asymptote

The horizontal asymptote of a function is a horizontal line that the graph approaches but never actually touches. In our given function, the horizontal asymptote is influenced by the vertical translation. Starting with the parent function \( f(x) = 2^x \), we observe that there's a vertical shift of 1 unit upwards due to the term \( 1 \). This results in the horizontal asymptote being \( y = 1 \). Essentially, as \( x \) approaches infinity or negative infinity, the function values get closer and closer to 1 but never actually reach it.

Domain and Range

The domain and range tell us the set of all possible input (x-values) and output (y-values) for our function, respectively.
The domain of an exponential function like \( f(x) = 1 - 2^{x+3} \) is all real numbers. Symbolically, we write this as \( (-\infty, +\infty) \). This means any real number can be plugged into the function without any restrictions.
For the range, we need to evaluate how the graph behaves vertically. Because of the vertical shift and reflection, our function can take any value less than 1 but never actually be 1. So, the range is \( (-\infty, 1) \). This tells us that \( y \) can be any number less than 1.

Vertical and Horizontal Transformations

Transformations are adjustments we make to the graph of the function to alter its shape or position.
**Horizontal Transformations:**

• The expression \( x + 3 \) indicates a horizontal shift. Since it’s \( x + 3\), the entire graph shifts \(-3 \) units to the left.

**Vertical Transformations:**

• The linear transformation \( -2^{x+3} \) includes a vertical reflection across the x-axis because of the negative sign.
• The term \(+1 \) means we then move the graph \(1 \) unit upwards.

Combining these gives the final transformed function: \ f(x) = 1 - 2^{x + 3}\.

Y-Intercept

The y-intercept is the point where the graph of the function crosses the y-axis. This happens when \( x = 0 \).
To find the y-intercept for \ f(x) = 1 - 2^{x+3}\, we substitute \( x = 0 \) into the function to get:
\