Gordon is evaluating some stocks he wants to purchase. You can see the function of the stock price below. The variable x represents days.

Stock A B C
Price f(x) = 25(1.08)x
x f(x)
0 22.00
1 21.56
2 21.13
The initial price of this stock is $30, but it has been increasing 4% each day.
Create the price function for Stock D. It should have the second lowest starting price and the second highest rate of growth. Using complete sentences, justify that your function meets these requirements.
Graph the price function for Stock D. Explain the key features of the graph with complete sentences.
A recent drop in sales has affected Stock D with the function g(x) = –6. Explain to Gordon how Stock D’s new price function, f(x) + g(x), will be created. Graph f(x) + g(x).
Gordon wants to fully understand what kind of changes could affect the money he is investing. Using complete sentences, explain the effect of the following transformations. Graph them and f(x) on the same coordinate plane.
f(x + 2)
f(–x)
f(x) + 3

To create the price function for Stock D, we need to find a function with the second lowest starting price and the second highest rate of growth, compared to the given stocks A, B, and C.

From the given data, we can see that Stock C has the lowest starting price of $21.13 on day 2, and Stock A has the highest rate of growth at 8% per day.

To have the second lowest starting price, Stock D's initial price needs to be higher than $21.13 but lower than $22.00. Let's choose $21.50 as the starting price.

To have the second highest rate of growth, Stock D's growth rate needs to be higher than 8% per day. Let's choose a growth rate of 12% per day.

Therefore, the price function for Stock D can be defined as f(x) = 21.50(1.12)^x.

To graph the price function for Stock D, we can plot the values of f(x) for various x values on a coordinate plane.

For example, if we substitute x = 0, 1, and 2 into the function, we get:

x = 0: f(0) = 21.50(1.12)^0 = 21.50
x = 1: f(1) = 21.50(1.12)^1 = 24.08
x = 2: f(2) = 21.50(1.12)^2 ≈ 27.04

Plotting these points, we have (0, 21.50), (1, 24.08), and (2, 27.04). The graph of the price function for Stock D increases exponentially with time, reflecting the compounding growth rate. As time increases, the rate of increase also accelerates.

Now, in response to the recent drop in sales, Stock D's new price function is given by f(x) + g(x), where g(x) represents the decrease in price due to the drop in sales.

Since g(x) = -6, we can add this constant value to each f(x) value to get the new price function.

For example, if we add -6 to each f(x) value we calculated earlier, we get:

x = 0: f(0) + g(0) = 21.50 - 6 = 15.50
x = 1: f(1) + g(1) = 24.08 - 6 = 18.08
x = 2: f(2) + g(2) ≈ 27.04 - 6 ≈ 21.04

Plotting these adjusted points, we can graph the price function f(x) + g(x) for Stock D.

To understand the effects of the transformations on the function f(x), we will graph them on the same coordinate plane.

1. Transformation f(x + 2):
This transformation represents a shift of the graph to the left by 2 units along the x-axis. This means that for each x value, we substitute x + 2 into the original function.

2. Transformation f(-x):
This transformation represents a reflection of the graph in the y-axis. It is achieved by substituting -x into the original function.

3. Transformation f(x) + 3:
This transformation represents a vertical shift of the graph upwards by 3 units along the y-axis. It is achieved by adding 3 to the f(x) values.

By graphing these transformations and the original function f(x) on the same coordinate plane, we can observe the changes in shape, position, and values of the graph.

To create the price function for Stock D, we need to find a function with the second lowest starting price and the second highest rate of growth. The given stocks have starting prices of $30, $22, $21.56, and $21.13. Among them, Stock C has the second lowest starting price of $21.56. Since it is not explicitly stated which stock has the second highest rate of growth, we can determine it by calculating the growth rate for each stock.

The growth rate for Stock A = (21.13 - 22) / (2 - 0) = -0.44
The growth rate for Stock B = (21.56 - 22) / (1 - 0) = -0.44
The growth rate for Stock C = (21.13 - 21.56) / (2 - 1) = -0.43

Among the given stocks, Stock C has the second highest rate of growth. Therefore, we will use the function of Stock C, which is f(x) = 25(1.08)^x, as the price function for Stock D.

To graph the price function for Stock D, we can plot the points provided in the table for Stock C and connect them with a smooth curve. The x-values will remain the same, from 0 to 2, and the y-values will be calculated using the function f(x) = 25(1.08)^x.

Here's a table for Stock D:

x f(x)
0 25
1 27
2 29.16

Features of the graph for Stock D:
1. The graph starts at a price of $25 on day 0.
2. The graph shows exponential growth as the days increase.
3. The rate of increase gradually accelerates over time due to the exponential nature of the function.

The recent drop in sales affected Stock D with the function g(x) = -6. To create the new price function for Stock D, we can simply add g(x) to the original price function f(x). Therefore, the new price function for Stock D is f(x) + g(x).

To graph f(x) + g(x), we need to add the corresponding values of f(x) and g(x) at each x-value and plot them. Here's a table representing the combined function:

x f(x) g(x) f(x) + g(x)
0 25 -6 19
1 27 -6 21
2 29.16 -6 23.16

By plotting the points (x, f(x) + g(x)), we will obtain the graph of f(x) + g(x) for Stock D.

Effects of transformations on the original function f(x):

1. f(x + 2): This transformation shifts the graph horizontally to the left by 2 units. Each point (x, f(x)) is shifted to the left by 2 units. The overall shape of the graph remains the same, but it is shifted to the left.

2. f(-x): This transformation reflects the graph across the y-axis. It means that for each point (x, f(x)), the x-coordinate is negated and the y-coordinate remains the same. The overall shape of the graph is flipped horizontally.

3. f(x) + 3: This transformation vertically shifts the graph upwards by 3 units. Each point (x, f(x)) is shifted upwards by 3 units. The overall shape of the graph remains the same, but it is shifted upwards.

Graphing these transformations along with the original graph f(x) on the same coordinate plane will provide a visual representation of the effect of each transformation.