Question

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

Short answer

The function \( f(x) = 3 \times \left( \frac{1}{2} \right)^x \) has a domain of \( (-\infty, \infty) \), a range of \( (0, \infty) \), a horizontal asymptote at \( y = 0 \), and a y-intercept at \((0, 3)\).

Step-by-step solution

Identify the Parent Function

The parent function is an exponential function of the form \( a \times b^x \). In this case, \( a = 3 \) and \( b = \frac{1}{2} \).

Graph the Parent Function

Graph the function \( y = \left(\frac{1}{2}\right)^x \). This is a decreasing exponential function passing through the points \((0,1)\), \((1, \frac{1}{2})\), etc.

Apply Vertical Stretch Transformation

Since the function is \( 3 \times \left( \frac{1}{2} \right)^x \), it includes a vertical stretch by a factor of 3. Multiply the y-coordinates of the points on the graph of the parent function by 3. For example, the point \((0,1)\) becomes \((0,3)\) and \((1, \frac{1}{2})\) becomes \((1, \frac{3}{2})\).

Plot the Transformed Function

Plot the new points from the previous step to get the graph of \( f(x) = 3 \times \left( \frac{1}{2} \right)^x \). The graph still passes through the y-intercept \((0, 3)\) and it decreases exponentially.

Determine the Domain

The domain of \( f(x) = 3 \times \left( \frac{1}{2} \right)^x \) is all real numbers. Thus, the domain is \( (-\infty, \infty) \).

Determine the Range

The range of the function is all positive real numbers since it never reaches zero and it extends to positive infinity as x approaches negative infinity. Thus, the range is \( (0, \infty) \).

Determine the Horizontal Asymptote

As x approaches positive infinity, \( f(x) = 3 \times \left( \frac{1}{2} \right)^x \) approaches zero, giving a horizontal asymptote at \( y = 0 \).

Find the y-Intercept

To find the y-intercept, set \( x = 0 \). Then, \( f(0) = 3 \times \left( \frac{1}{2} \right)^0 = 3 \). So the y-intercept is \((0, 3)\).

Key concepts

transformations in functions

When graphing functions, transformations help modify their shapes and positions. For the given function:

$$ f(x) = 3 \times \bigg( \frac{1}{2} \bigg)^x $$

First, recognize the parent function, which is an exponential decay function:

$$ y = \bigg( \frac{1}{2} \bigg)^x $$

This function decreases as x increases. Key points to plot include

• (0,1)
• (1, \frac{1}{2})

Since our given function has a coefficient of 3, it applies a vertical stretch by a factor of 3. This changes the y-coordinate values of the points we plot. Instead of plotting (0,1), we now plot (0,3), and instead of plotting (1, \frac{1}{2} ), we plot (1, \frac{3}{2} ). After applying these transformations, you get the transformed graph.

domain and range

The domain and range define the possible values of x and y that the function can take.

For the function:
$$ f(x) = 3 \times \bigg( \frac{1}{2} \bigg)^x $$
The domain is all real numbers. This means:

• $$ (-dy\to)$$
• ∞)$$

The range of the function, or the set of possible y-values, only includes positive numbers because the function never touches or goes below zero. Specifically, the range is

• $$ (0,∞)$$

Functions of this form will always have these ranges for exponential decay functions, multiplied by positive constants.

horizontal asymptote

A horizontal asymptote is a horizontal line that the graph approaches but never touches. For exponential functions like:
$$ f(x) = 3 \times \bigg( \frac{1}{2} \bigg)^x $$
The horizontal asymptote is y = 0.
As x increases to positive infinity, the value of $$ ( \frac{1}{2})^x $$ becomes extremely small.
Multiplying by 3 does not affect the limit. Hence, the function approaches zero but never reaches it.

y-intercept

The y-intercept is where the graph crosses the y-axis. For this, set x to 0 in the function:
$$ f(0) = 3 \times \bigg( \frac{1}{2} \bigg)^0 $$
Since any number to the power of 0 is 1, this simplifies to:
$$ f(0) = 3 \times 1 = 3 $$
So, the y-intercept of the function is:
$$ (0, 3) $$
This is where the graph will cross the y-coordinate.