Question

use technology to find points and then graph the function $y = 2^x - 4$, following the instructions below. equation of asymptote: $y = -4$ 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 (any integer x will work, we'll compute y for these).

Step2: Compute y for x = -2

Substitute x = -2 into \( y = 2^x - 4 \).
\( 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \), so \( y = \frac{1}{4} - 4 = -\frac{15}{4} = -3.75 \). Wait, not integer. Let's try x = 0.

Step3: Compute y for x = 0

\( y = 2^0 - 4 = 1 - 4 = -3 \). So point (0, -3).

Step4: Compute y for x = 1

\( y = 2^1 - 4 = 2 - 4 = -2 \). Point (1, -2).

Step5: Compute y for x = 2

\( y = 2^2 - 4 = 4 - 4 = 0 \). Point (2, 0).

Step6: Compute y for x = 3

\( y = 2^3 - 4 = 8 - 4 = 4 \). Point (3, 4).

Step7: Check x = -1

\( y = 2^{-1} - 4 = \frac{1}{2} - 4 = -3.5 \). Not integer. x = -2 gave non-integer, x = 0,1,2,3 give integer y. So four points: (0, -3), (1, -2), (2, 0), (3, 4). Also, x = 4: \( y = 2^4 - 4 = 16 - 4 = 12 \), but (3,4) is simpler. Alternatively, x = -1 non-integer, x = 0,1,2,3 work.

Answer:

Four points with integer coordinates: (0, -3), (1, -2), (2, 0), (3, 4) (plot these on the graph). The asymptote is \( y = -4 \).