Answer:
[tex]y=e^{x-2}+9[/tex]
Step-by-step explanation:
The parent exponential function is:
[tex]y = e^x[/tex]
The graph of the parent function has the following characteristics:
- The function is strictly increasing, meaning as x increases, the corresponding values of y also increase.
- The domain is all real numbers (-∞, ∞) and the range is (0, ∞).
- As x approaches negative infinity, the parent function approaches but never reaches the x-axis (y = 0). So, there is a horizontal asymptote at y = 0.
- The function passes through the point (0, 1), so the y-intercept is y = 1.
Given that the graphed function has not been stretched or compressed, this implies that we don't need to multiply the parent function or its independent variable by a scale factor.
The range of the graphed function is (9, ∞). This suggests a vertical translation of the parent function by 9 units upwards. Therefore, we add 9 to the parent function:
[tex]y = e^x + 9[/tex]
Considering that the parent function crosses the y-axis at (0, 1), a translation of 9 units upward would place it crossing the y-axis at (0, 10). However, the graphed function does not intersect the y-axis at y = 10, indicating the need for a horizontal translation.
To determine the nature of the horizontal translation, we need to find the value of x when y = 10. Since x = 2 when y = 10, it indicates that the graphed function has undergone a translation of 2 units to the right. When we translate a function n units to the right, we subtract n from the independent variable. Therefore:
[tex]y = e^{x-2} + 9[/tex]
So, the equation of the graph is:
[tex]\Large\boxed{\boxed{y = e^{x-2} + 9}}[/tex]
This is a translation of the parent function by 2 units to the right and 9 units up.
