Question

The equation for an exponential function is p(x) = 4 · (0.75)x – 3. Graph the function.

Answer 1

The graph of p(x) = 4 * (0.75)^x - 3 starts at (0, 1), exhibits exponential decay, and approaches the line y = -3 as x increases.

The given exponential function is p(x) = 4 * (0.75)^x - 3. To graph this function, we can observe its key features. The base of the exponential term is 0.75, which is between 0 and 1, indicating decay. The multiplication by 4 scales the function vertically, and the subtraction of 3 shifts it downward.

Starting with the initial point when x = 0, p(0) = 4 * (0.75)^0 - 3 = 4 - 3 = 1. This gives us the point (0, 1) on the graph.

As x increases, the exponential term decreases, leading to a decrease in p(x). The graph will approach, but never touch, the horizontal line y = -3 due to the subtraction of 3 in the function.

To graph, plot additional points by selecting various x values and calculating p(x). As x becomes more negative, p(x) approaches zero, resulting in an asymptote at y = -3.

In summary, the graph of the exponential function p(x) = 4 * (0.75)^x - 3 starts at (0, 1), decays exponentially as x increases, and approaches the horizontal line y = -3.

Answer 2

Explanation:

To graph the exponential function p(x) = 4 * (0.75)^x - 3, you can follow these steps:

1. Choose a range of x-values: Decide on a range of x-values that you want to plot on the graph. It's a good practice to select both positive and negative values to get a complete view of the function's behavior. For example, you can choose x-values from -5 to 5.

2. Calculate corresponding y-values: For each x-value in your chosen range, calculate the corresponding y-value using the function p(x). Plug each x-value into the function and compute p(x).

3. Plot the points: Create a set of ordered pairs (x, p(x)) and plot them on a graph. Use a coordinate plane with x-axis and y-axis labeled with appropriate scales.

4. Connect the points: Connect the plotted points smoothly. An exponential function typically exhibits exponential growth or decay, so the curve should either rise or fall continuously.

Here's a table of values and a corresponding graph for p(x) = 4 * (0.75)^x - 3:

| x | p(x) |

|------|-----------------------|

| -5 | 5.15625 |

| -4 | 6.84375 |

| -3 | 8.625 |

| -2 | 10.875 |

| -1 | 13.5 |

| 0 | 16 |

| 1 | 20 |

| 2 | 25 |

| 3 | 31.25 |

| 4 | 39.0625 |

| 5 | 48.828125 |

Now, plot these points on a graph, and you should see an exponential decay curve. Be sure to label your axes and indicate the behavior of the curve as it approaches the x-axis.

The graph will show that the function p(x) decreases rapidly as x increases, but it never quite reaches zero.