graph the function.\nh(x)=8cdot(\frac{3}{4})^{x}

Explanation:

Step 1: Identify the function type and base

Exponential function with base ( \frac{3}{4} ) (between 0 and 1, so decreasing).

Step 2: Determine the y-intercept

When ( x = 0 ), ( h(0) = 8 \cdot \left( \frac{3}{4} \right)^0 = 8 \cdot 1 = 8 ), so the y-intercept is ((0, 8)).

Step 3: Analyze end behavior

As ( x \to \infty ), ( \left( \frac{3}{4} \right)^x \to 0 ), so ( h(x) \to 0 ) (horizontal asymptote at ( y = 0 )).
As ( x \to -\infty ), ( \left( \frac{3}{4} \right)^x \to \infty ), so ( h(x) \to \infty ).

Step 4: Plot key points

At ( x = 1 ): ( h(1) = 8 \cdot \frac{3}{4} = 6 ) → point ((1, 6)).
At ( x = -1 ): ( h(-1) = 8 \cdot \frac{4}{3} = \frac{32}{3} \approx 10.67 ) → point ((-1, \frac{32}{3})).

Answer:

The graph is a decreasing exponential curve passing through ((0, 8)) and ((1, 6)), approaching the x-axis ((y = 0)) as (x) increases and rising to infinity as (x) decreases.