Problem 43 - Precalculus Mathematics for Calculus

Question

Draw graphs of the given family of functions for \(c=0.25,0.5,1,2,4 .\) How are the graphs related? $$f(x)=c 2^{x}$$

Step-by-Step Solution

Answer

Graphs are vertically stretched versions of each other, steeper for larger \( c \).

Step 1: Understanding the Function

The given family of functions is described by the equation \( f(x) = c \cdot 2^x \). The variable \( c \) is a constant that changes the steepness of the graph. Our task is to draw the graphs of this function for different values of \( c \) (0.25, 0.5, 1, 2, and 4) and then determine how the graphs are related.

Step 2: Calculating Points for Graphing

For each value of \( c \), we calculate several points. Use \( x = -2, -1, 0, 1, \) and \( 2 \), and substitute into the function \( f(x) = c \cdot 2^x \). This will create points to plot the graph. For example, if \( c = 1 \), the points would be \((-2, 0.25), (-1, 0.5), (0, 1), (1, 2), (2, 4)\). Repeat this for each \( c \) value.

Step 3: Plotting the Graphs

On the same coordinate system, plot the points calculated for each value of \( c \) and connect them to create a smooth curve. Each value of \( c \) will result in a different exponential graph, all passing through the origin when \( x = 0 \) at the point \( (0,c) \).

Step 4: Analyzing the Relationship

Observe how the value of \( c \) affects the graphs. Increasing values of \( c \) results in graphs that are steeper and increase faster. All graphs pass through the point \( (0, c) \), and for any given value of \( x \), larger \( c \) results in a higher \( y \)-value.

Step 5: Conclusion

The graphs are vertically stretched versions of each other depending on \( c \). If \( c \) is larger, the graph is steeper, indicating a faster rate of growth for the exponential function.

Key Concepts

Graphical Representation

When working with exponential functions like \( f(x) = c \cdot 2^x \), it's helpful to visualize how they appear on a graph. The graphical representation shows how the function behaves and changes for different values of \( c \).
In this case, we're plotting graphs for \( c = 0.25, 0.5, 1, 2, \) and \( 4 \). By calculating function values for a set of \( x \) values and plotting them, you'll notice the shape and steepness of the curve changes.
Key points to consider when plotting:

• The graph will always pass through the point \((0, c)\) because when \( x = 0 \), \( 2^0 = 1 \) and so \( f(x) = c \cdot 1 = c \).
• For negative \( x \)-values, the function gets closer to zero but never actually reaches it.
• The curve gets steeper as you increase \( x \) for a given \( c \). Higher \( c \) values also increase the steepness.

Observing the graph can provide insight into how exponential functions grow and can be applied in various fields such as compound interest calculations, population growth models, and more.

Function Transformation

Function transformation is a crucial concept to understand when examining the effect of changing \( c \) in our exponential function \( f(x) = c \cdot 2^x \). Connecting this with the idea of a 'family of functions', we can see that different values of \( c \) result in related graphs, with each being a transformed version of the others.
There are a few types of transformations you might encounter:

• Vertical Translation: This occurs when we add or subtract a constant, but in \( f(x) = c \cdot 2^x \) there is no explicit vertical translation unless the expression is modified further.
• Vertical Stretch/Compression: Here, the function changes based on the constant \( c \). If \( c > 1 \), the graph stretches upwards, away from the x-axis. For \( 0 < c < 1 \), it compresses towards the x-axis.
• Reflection: Negative values would reflect the graph across the x-axis, but since \( c \) is positive, we don't see this effect here.

Understanding these transformations helps identify how the graphs are connected and allows for predictions about function behavior without needing to plot every single point.

Vertical Stretch

The concept of vertical stretch is seen clearly in the function \( f(x) = c \cdot 2^x \). This transformation describes changes in the graph's slope and steepness. Here's how it works:
- When \( c > 1 \), the graph experiences a vertical stretch, pulling it further from the x-axis. This makes the graph steeper, indicating a faster rate of growth.
- If \( 0 < c < 1 \), the graph is compressed, which makes it less steep.
- Each graph in our set passes through the point \((0, c)\), and for larger \( c \), the graph climbs more swiftly as \( x \) increases.
This concept is vital in fields like physics and engineering where exponential growth or decay models must be visualized. In practical situations, knowing how changing a parameter affects a graph's shape can lead to better design and analysis of systems.