Problem 50 - Algebra 1: Concepts and Skills

Question

Use a calculator to investigate the effects of a and b on the graph of \(y=a b^{x}\) In the same viewing rectangle, graph \(y=2(2)^{x}, y=4(2)^{x},\) and \(y=8(2)^{x}\) How does an increase in the value of \(a\) affect the graph of \(y=a b^{x} ?\)

Step-by-Step Solution

Answer

Increasing the \(a\) value in the function \(y = a b^x\) leads to a vertical stretching of the graph. The basic shape of the graph is retained, but the height increases.

Step 1: Graph the first function

First, start by making a graph of the function \(y = 2(2^x)\). This is the baseline function used for comparison. The \(a\) value in this function is 2.

Step 2: Graph the second function

Next, graph the second function \(y=4(2)^x\). Here the \(a\) value has been doubled compared to the previous function and we will observe the effect this has on the graph.

Step 3: Graph the third function

Now, graph the third function \(y=8(2)^x\). Here the \(a\) value is four times larger than the original function. We will then see how this further increase affect the graph.

Step 4: Compare the Graphs

Examine the three graphs. Be particularly focused on understanding how the shape and position of the curve changes as the \(a\) value increases. Notice that an increase in the \(a\) value leads to a vertical stretching of the graph, meaning that the graph gets higher but retains its basic shape.

Key Concepts

Graph Transformations

Graph transformations involve altering the position or shape of a graph based on changes to the function's parameters. This specific exercise focuses on the exponential function, which typically has the form \( y = ab^x \). In this general form:

• \( a \) is a coefficient affecting the vertical stretch of the graph.
• \( b \) is the base of the exponential function and affects how steep or flat the graph is.

Remember, graph transformations can include translations (shifting up, down, left, or right), reflections (flipping the graph over an axis), or scalings (stretching or compressing the graph). Here, we'll focus on how changing the coefficient \( a \) transforms an exponential graph vertically, impacting its height and position on the coordinate plane.
Exploring graph transformations visually with a calculator can help you see these changes in real time. This enables a deeper understanding of the abstract mathematical concepts as they apply to exponential functions.

Effect of Coefficient on Graphs

The coefficient \( a \) in an exponential function \( y = ab^x \) significantly influences the graph. As seen in this exercise, different values of \( a \) lead to different shapes of the graph.

• An increase in \( a \) makes the graph rise more steeply.
• Decreasing \( a \) would compress the graph towards the x-axis.

Changing this coefficient essentially scales every point on the graph. But it doesn’t change the growth rate, which is determined by \( b \).
For example, in the given functions \( y=2(2)^x \), \( y=4(2)^x \), and \( y=8(2)^x \), you will see that as \( a \) increases (2, 4, 8), the height of the graph in the same x-value becomes larger. Despite these changes, each graph maintains the same exponential growth pattern, illustrating that the basic structure of the curve is retained even as \( a \) varies.

Function Graphing

Graphing functions, especially exponential ones, is a visual activity that can enhance understanding. By plotting these functions on the same set of axes, you can easily compare how they behave differently. Here are some steps to consider when graphing:

• Determine the effect of each parameter individually while keeping the others constant.
• Use a consistent scale on your axes to make comparisons clearer.
• Watch how the graph changes at key points (like where the function intersects the axes).

When graphing a complex function like the exponential form \( y = ab^x \), start with a simpler base function like \( y = 2^x \), and then apply the transformations step by step. As seen in this exercise, changing the coefficient \( a \) modifies the graph vertically. Practicing with graphing tools or calculators can greatly improve your intuition for how exponential functions behave.

Vertical Stretching in Graphs

Vertical stretching involves increasing or decreasing the distance between points on a graph and the x-axis. Specifically, for exponential functions \( y = ab^x \), changing the coefficient \( a \) results in a vertical stretch.

• A larger \( a \) value pulls the graph upward, making it taller at every point.
• A smaller \( a \) value pushes the graph downward, making it shorter.

As you increase \( a \), you will notice that for a given x-value, the value of \( y \) is proportionately larger. This does not alter the exponential nature of the curve, but instead amplifies or reduces its height.
This concept is critical for understanding how different factors affect real-world phenomena like population growth or interest in financial calculations. Comprehensive graph analysis aids in visualizing how swiftly these changes occur as parameters are modified.