Problem 10 - Algebra A Combined Function

Question

Graph each exponential function. $$ y=-3^{x} $$

Step-by-Step Solution

Answer

Graph reflects across x-axis and shows exponential decay.

Step 1: Identify the Base Function

The base function here is an exponential function of the form \( y = a^x \), where \( a = 3 \). This indicates that the basic form of the function is exponential growth by a factor of 3 for every increase in \( x \).

Step 2: Recognize the Transformation

In the given function \( y = -3^x \), the negative sign in front of \( 3^x \) indicates a reflection across the x-axis. This means instead of growing upwards as \( x \) increases, the function will decrease.

Step 3: Create a Table of Values

Choose a few values of \( x \) to substitute into the function \( y = -3^x \), such as \( x = -1, 0, 1, 2 \). Calculate the corresponding \( y \) values: - For \( x = -1 \), \( y = -\frac{1}{3} \)- For \( x = 0 \), \( y = -1 \)- For \( x = 1 \), \( y = -3 \)- For \( x = 2 \), \( y = -9 \)

Step 4: Plot the Points and Draw the Graph

Using the calculated points \((-1, -\frac{1}{3})\), \((0, -1)\), \((1, -3)\), and \((2, -9)\), plot these on a coordinate grid. Connect the points with a smooth curve, showing the exponential decay and reflection over the x-axis.

Step 5: Analyze the Graph

Notice that as \( x \) increases, the values of \( y \) become more negative, confirming that the function is decreasing due to the negative sign. The graph will approach but never touch the x-axis, which acts as a horizontal asymptote.

Key Concepts

Graphical Representation

Visualizing an exponential function can be quite insightful. For the function \( y = -3^x \), it is crucial to interpret its graphical attributes. When we graph this function, we see its unique behavior due to the negative sign preceding the \( 3^x \). This changes how the graph appears compared to standard exponential growth functions.

• The function follows a specifically mirrored pattern across the x-axis.
• This happens because each calculated \( y \) value is multiplied by -1.

To graph \( y = -3^x \), start by plotting a few points on the coordinate plane, like \((-1, -\frac{1}{3})\), \((0, -1)\), \((1, -3)\), and \((2, -9)\). These points are then connected using a smooth curve. Notice how the curve stretches downward as \( x \) increases. Remember, graphical representation emphasizes not just where a function starts or stops, but how it behaves as a whole.

Exponential Decay

The notion of exponential decay is central to understanding how the function \( y = -3^x \) behaves. Normally, exponential functions grow, showcasing exponential growth. However, the negative sign introduces decay rather than growth.Here's what happens:

• With exponential decay, the function values decrease rapidly.
• As \( x \) increases, \( y \) values drop fast but never hit zero.

This fast drop aligns with the idea that at every step, the function loses magnitude exponentially. In our case, the decay occurs downwards on the graph due to the negative sign. It highlights how negative transformations affect exponential functions, converting typical growth patterns into decaying curves.

Function Transformation

Transformations are crucial in shifting a function's graph. The function \( y = -3^x \) illustrates this well with its transformation from a basic exponential function. Let's explore these transformations:

• The negative sign in \( -3^x \) creates a vertical reflection.
• This transformation flips the graph over the x-axis.

Such reflections might initially seem subtle, but they drastically alter the function's output for each \( x \) value. In practical terms, every positive \( 3^x \) value turns negative, flipping the entire curve's orientation. Function transformations like these help us appreciate how small algebraic adjustments significantly impact a function's graphical output. Understanding transformations expands your grasp on how functions behave and how to predict and visualize changes.