Question

1. sketch the graph of an exponential function using transformations and find its domain, range and horizontal asymptote- week 13 - (section 4.1)

• ( y = -3^{-0.5x} + 2 )

Explanation:

Step1: Analyze the parent function

The parent exponential function is \( y = 3^x \). Its domain is \( (-\infty, \infty) \), range is \( (0, \infty) \), and horizontal asymptote is \( y = 0 \).

Step2: Apply horizontal stretch/compression

For \( y = 3^{-0.5x} \), the exponent has a coefficient of \(-0.5\). The horizontal stretch factor is \( \frac{1}{| - 0.5|}=2 \), and the negative sign reflects the graph over the \( y \)-axis. So \( y = 3^{-0.5x} \) is a horizontal stretch by a factor of 2 and reflection over the \( y \)-axis of \( y = 3^x \). Its domain is still \( (-\infty, \infty) \), range \( (0, \infty) \), horizontal asymptote \( y = 0 \).

Step3: Apply vertical reflection

For \( y=-3^{-0.5x} \), the negative sign reflects the graph of \( y = 3^{-0.5x} \) over the \( x \)-axis. Now the range becomes \( (-\infty, 0) \), domain \( (-\infty, \infty) \), horizontal asymptote \( y = 0 \).

Step4: Apply vertical shift

For \( y=-3^{-0.5x}+2 \), we shift the graph of \( y=-3^{-0.5x} \) up by 2 units.

Domain:

Exponential functions with real - valued exponents have a domain of all real numbers. So the domain of \( y=-3^{-0.5x}+2 \) is \( (-\infty, \infty) \).

Range:

The range of \( y=-3^{-0.5x} \) is \( (-\infty, 0) \). When we shift it up by 2 units, we add 2 to each \( y \) - value. So the range of \( y=-3^{-0.5x}+2 \) is \( (-\infty, 2) \).

Horizontal Asymptote:

The horizontal asymptote of \( y=-3^{-0.5x} \) is \( y = 0 \). After shifting up by 2 units, the horizontal asymptote becomes \( y=2 \).

Sketching the graph:

1. Start with the parent function \( y = 3^x \) (increasing, passes through \( (0,1) \)).
2. Reflect over the \( y \)-axis and stretch horizontally by a factor of 2 to get \( y = 3^{-0.5x} \) (decreasing, passes through \( (0,1) \)).
3. Reflect over the \( x \)-axis to get \( y=-3^{-0.5x} \) (increasing, passes through \( (0, - 1) \)).
4. Shift up by 2 units to get \( y=-3^{-0.5x}+2 \) (increasing, passes through \( (0,1) \)).

Answer:

• Domain: \( (-\infty, \infty) \)
• Range: \( (-\infty, 2) \)
• Horizontal Asymptote: \( y = 2 \)