Question

use technology to find points and then graph the function $y = 2^x - 2$, following the instructions below.
equation of asymptote:
$y = -2$
plot at least four points with integer coordinates that fit on the axes below. click a point to delete it.

Explanation:

Step1: Choose integer x-values

Let's pick \( x = -2, -1, 0, 1, 2, 3 \) (we need at least four, so we can use these).

Step2: Calculate y for x = -2

Substitute \( x = -2 \) into \( y = 2^x - 2 \):
\( y = 2^{-2} - 2 = \frac{1}{4} - 2 = -\frac{7}{4} \) (not integer, so skip).

Step3: Calculate y for x = -1

Substitute \( x = -1 \):
\( y = 2^{-1} - 2 = \frac{1}{2} - 2 = -\frac{3}{2} \) (not integer, skip).

Step4: Calculate y for x = 0

Substitute \( x = 0 \):
\( y = 2^0 - 2 = 1 - 2 = -1 \). So point: \( (0, -1) \).

Step5: Calculate y for x = 1

Substitute \( x = 1 \):
\( y = 2^1 - 2 = 2 - 2 = 0 \). Point: \( (1, 0) \).

Step6: Calculate y for x = 2

Substitute \( x = 2 \):
\( y = 2^2 - 2 = 4 - 2 = 2 \). Point: \( (2, 2) \).

Step7: Calculate y for x = 3

Substitute \( x = 3 \):
\( y = 2^3 - 2 = 8 - 2 = 6 \). Point: \( (3, 6) \).

Step8: Verify asymptote

The function \( y = 2^x - 2 \) is an exponential function. The parent function \( y = 2^x \) has horizontal asymptote \( y = 0 \). Shifting down 2 units gives asymptote \( y = -2 \), which matches the given boxed value.

Answer:

Asymptote: \( y = -2 \)
Points (e.g.): \( (0, -1) \), \( (1, 0) \), \( (2, 2) \), \( (3, 6) \) (plot these on the graph).