Problem 3 - Algebra A Combined Function

Question

Graph each exponential function. $$ y=1+2^{x} $$

Step-by-Step Solution

Answer

Plot \((-2, 1.25), (-1, 1.5), (0, 2), (1, 3), (2, 5)\) and draw the curve through them.

Step 1: Understand the Exponential Function

The function given is \(y = 1 + 2^x\). This is an exponential function, where \(2^x\) is the exponential part and the function is shifted vertically up by 1 unit due to the +1.

Step 2: Create a Table of Values

To graph the function, compute some values of \(y\) for chosen values of \(x\). For example, try \(x = -2, -1, 0, 1,\) and \(2\). Calculate the corresponding \(y\) values.- When \(x = -2\), \(y = 1 + \frac{1}{4} = 1.25\)- When \(x = -1\), \(y = 1 + \frac{1}{2} = 1.5\)- When \(x = 0\), \(y = 1 + 1 = 2\)- When \(x = 1\), \(y = 1 + 2 = 3\)- When \(x = 2\), \(y = 1 + 4 = 5\)

Step 3: Plot the Points

Use the calculated values to plot the points on a coordinate grid. The points to plot are \((-2, 1.25), (-1, 1.5), (0, 2), (1, 3), (2, 5)\).

Step 4: Draw the Exponential Curve

After plotting the points, draw a smooth curve through these points. The curve should show an upward trend as \(x\) increases, reflecting the exponential growth of the function.

Key Concepts

graphing

Graphing an exponential function is a visual way of understanding how the function behaves as the input values change. In the function given, \( y = 1 + 2^x \), we start by evaluating this expression for specific values of \( x \). By doing this, we generate a set of \( (x, y) \) coordinates that reflect the arithmetic computations of the function.

In this case, the function encompasses both the exponential term \( 2^x \) and a constant shift in the vertical direction by +1, indicating it will not pass through the origin but will instead shift all function values upwards by 1 unit. After plotting these points on the graph, you draw a curve to connect them, which will highlight the dynamic behavior of exponential functions. This step visually represents the continuous increase, emphasizing the steep incline associated with exponential growth.

coordinate grid

The coordinate grid is an essential tool for plotting the calculated points of an exponential function. This grid consists of two axes: the horizontal \( x \)-axis and the vertical \( y \)-axis. Each axis provides a scale to measure distances and values for the input (\( x \)) and output (\( y \)) of the function.

When using a coordinate grid, the center or the 'origin' is labeled as \( (0,0) \). Points are plotted in relation to their \( x \) and \( y \) coordinates. For the function \( y = 1 + 2^x \), the points \((-2, 1.25), (-1, 1.5), (0, 2), (1, 3), (2, 5)\) help visualize the function.

• Each point showcases a specific value of \( x \) leading to a calculated value of \( y \).
• Marking these points accurately on the grid provides a clear representation of the function's growth pattern.

Once the points are plotted, the grid helps in drawing a smooth curve to represent the function.

transformations

Transformations alter the appearance and location of a basic function on a graph. For the exponential function \( y = 1 + 2^x \), transformations play a pivotal role in understanding its graph.

In this function, the primary transformation is the vertical shift caused by '+1'. This means every point on the basic curve \( 2^x \) is shifted one unit upwards. This shift is significant because it changes the starting point on the \( y\)-axis without altering the rate of exponential growth.

• Vertical shifts: Add or subtract numbers directly to the exponential function. Here, +1 moves the entire graph up.
• Horizontal shifts: These involve changes in the input \( x \), but are not present in our exercise.
• Scaling transformations can stretch or compress the graph, often by multiplying the exponential component. In \( y = 1 + 2^x \), there is no horizontal shift or scaling, just the simple vertical translation.

Understanding transformations is essential to accurately predict how the graph will look and where it will be positioned.

exponential growth

Exponential growth is characterized by a quick increase in values, which is one of the defining traits of exponential functions. In the given equation \( y = 1 + 2^x \), the function grows exponentially because it contains the term \( 2^x \).

Exponential growth means as \( x \) increases, \( y \) increases at an accelerating rate—doubling with each increment of \( x \). This results in a graph that starts slowly at lower values of \( x \) and steepens sharply at higher \( x \) values.

• Initial slow growth occurs at negative and small positive \( x \) values, shown by points like \( (-2, 1.25) \) and \( (0, 2) \).
• The rate of growth becomes rapidly steep, as seen in points like \( (2, 5) \).

This behavior is what makes exponential functions fundamental in modeling real-world scenarios such as population growth, financial investments, and certain physics phenomena.