Answer:
y-intercept = (0, 2.02)
As x → +∞, y → +∞
As x → -∞, y → 2⁺
Growth function
Step-by-step explanation:
Given exponential function:
[tex]y=\dfrac{1}{2}(25)^{x-1}+2[/tex]
The y-intercept is the point at which the graph of the function intersects the y-axis, so when x = 0. Therefore, to find the y-intercept, substitute x = 0 into the function:
[tex]y=\dfrac{1}{2}(25)^{0-1}+2 \\\\\\ y=\dfrac{1}{2}(25)^{-1}+2 \\\\\\ y=\dfrac{1}{2}\cdot \dfrac{1}{25}+2 \\\\\\ y=\dfrac{1}{50}+2\\\\\\y=2.02[/tex]
So, the y-intercept is (0, 2.02).
The end behavior of an exponential function depends on the base of the exponent and the sign of the function. In this case, the base is 25, and the function is positive.
When the base is greater than 1 and the function is positive, the function represents exponential growth. So, as x approaches positive infinity, the function will grow without bound.
Since the function has 2 added to it, it will shift the entire graph vertically upwards by 2 units. This means that as x approaches negative infinity, the function will approach y = 2 (from the positive side) but never touch it.
Therefore, the end behaviors of the function are:
- As x → +∞, y → +∞.
- As x → -∞, y → 2⁺.
