Problem 11 - Algebra 2

Question

Graph each function as a transformation of its parent function. $$ y=-(0.3)^{x-2} $$

Step-by-Step Solution

Answer

The transformed function \(y=-(0.3)^{x-2}\) is obtained from the parent function \(y=b^x\) by applying a horizontal shift to the right by 2 units, reflecting vertically about the \(x\)-axis, and vertically scaling by a factor of 0.3.

Step 1: Identify the Parent Function and its Transformations

The parent function, in this case, is \(y=b^x\), where \(b > 0\) and \(b ≠ 1\). This is an exponential function. The given function has three transformations: 1) a vertical reflection indicated by the negative sign in front of the expression, 2) a horizontal shift due to \(x-2\) in place of \(x\), and 3) vertical dilation because of the multiplication by 0.3.

Step 2: Graph the Parent Function

Draw the graph of the parent function \(y=b^x\), where \(b=0.3\). This provides a basis for graphing the transformed function. The graph will decrease as \(x\) increases because 0.3 is between 0 and 1.

Step 3: Shift the Function Horizontally

The exponent is \(x-2\) instead of \(x\), indicating a horizontal shift. It shifts two units to the right, so every point on the graph of the parent function is moved two units to the right.

Step 4: Reflect the Function Vertically

The negative sign in front of the expression indicates a vertical reflection around the \(x\)-axis, so flip the graph around the \(x\)-axis.

Step 5: Scale the Function Vertically

The term \(0.3^x\) implies a vertical dilation by a factor of 0.3. This means the transformed function graph would appear narrower than the original graph.

Key Concepts

Exponential Function

Exponential functions are a special type of mathematical function where the variable, often represented as \( x \), is the exponent of a constant base. The general form of an exponential function can be written as \( y = b^x \), where \( b \) is a positive constant different from one. If the base \( b \) is greater than 1, the exponential function represents exponential growth, meaning that as \( x \) increases, \( y \) increases rapidly. Conversely, if \( 0 < b < 1 \), the function represents exponential decay, where the value of \( y \) decreases as \( x \) increases.

In our original exercise, the function \( y = -(0.3)^{x-2} \) demonstrates exponential decay because the base \( 0.3 \) is less than 1. This makes it a very good example of how exponential functions can behave differently based on the value of \( b \). Understanding the nature of exponential functions is crucial as they appear in many real-life scenarios such as population growth, radioactive decay, and interest calculations in finance.

Parent Function

The concept of a parent function is essential in understanding transformations. In mathematics, a parent function is the simplest form of a function type. It serves as the foundation upon which transformations occur. For exponential functions, the parent function is usually \( y = b^x \), where you have a constant base \( b \).

In our exercise, the parent function \( y = (0.3)^x \) is transformed through various operations to yield the function \( y = -(0.3)^{x-2} \). By knowing the shape and behavior of the parent function, we can visualize how the graph changes with different transformations.

• The basic graph of a parent exponential function \( y = b^x \) is a curve that increases or decreases exponentially, depending on the value of \( b \).
• Transformations such as shifts, reflections, and dilations are applied to this parent function to graph new functions.

Recognizing the parent function allows you to better predict how a graph will transform, making it an essential concept for graphing transformations.

Graphing Transformations

Graphing transformations involves changing the position and shape of a graph based on certain algebraic alterations. When graphing the function \( y = -(0.3)^{x-2} \), consider how these transformations affect the graph.

1. **Vertical Reflection:** The negative sign in front of the base causes a vertical reflection. This flips the graph over the \( x \)-axis, transforming the original exponential decay graph into one that decays in the negative direction.

2. **Horizontal Shift:** The \( x - 2 \) part indicates a horizontal shift of the graph. Instead of starting at \( x = 0 \), the graph shifts two units to the right because the subtraction inside the exponent delays the function's actions by two units.

3. **Vertical Dilation:** The base \( 0.3 \) inherently provides a vertical dilation. With a smaller base, the graph appears compressed or narrowed, emphasizing the rapid approach towards the axis as \( x \) increases.

• All these transformations combine to produce the final graph.
• Understanding each step helps you sketch graphs accurately and predict the effects of each transformation.

Graphing transformations not only involves shifting curves but also altering their appearance by reflecting and dilating them, giving a complete perspective on how diverse functions can be manipulated.