Determine the coordinates of A (y-intercept).

Question

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Here are the solutions to the problems based on the given exponential function $y = \left(\frac{1}{2}\right)^x - 4$ .

Determine the coordinates of A (y-intercept).
The y-intercept occurs when $x=0$ .
Substitute $x=0$ into the equation:
$y = \left(\frac{1}{2}\right)^0 - 4$ $y = 1 - 4$ $y = -3$
The coordinates of A are $\boxed{(0, -3)}$ .
Determine the coordinates of B (x-intercept).
The x-intercept occurs when $y=0$ .
Substitute $y=0$ into the equation:
$0 = \left(\frac{1}{2}\right)^x - 4$ $4 = \left(\frac{1}{2}\right)^x$
Rewrite 4 as a power of 2 and $\frac{1}{2}$ as a power of 2:
$2^2 = (2^{-1})^x$ $2^2 = 2^{-x}$
Equate the exponents:
$2 = -x$ $x = -2$
The coordinates of B are $\boxed{(-2, 0)}$ .
Equation of the asymptote of y.
For an exponential function of the form $y = ab^x + c$ , the horizontal asymptote is $y=c$ .
In the given equation $y = \left(\frac{1}{2}\right)^x - 4$ , the value of $c$ is $-4$ .
The equation of the asymptote is $\boxed{y = -4}$ .
Domain and range of y.
The domain of an exponential function is all real numbers.
The range is determined by the horizontal asymptote. Since the base $\frac{1}{2}$ is between 0 and 1, the function decreases and approaches the asymptote $y=-4$ from above.
The domain is $\boxed{x \in \mathbb{R}}$ .
The range is $\boxed{y > -4}$ .

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