Question

use technology to find points and then graph the function $y = 2^x - 7$, following the instructions below.
plot at least five points with integer coordinates that fit on the axes below. click a point to delete it.
done plotting points

Explanation:

Step1: Choose x-values

Let's pick integer values for \( x \), such as \( x = -2, -1, 0, 1, 2 \).

Step2: Calculate y for x=-2

Substitute \( x = -2 \) into \( y = 2^x - 7 \):
\( y = 2^{-2} - 7 = \frac{1}{4} - 7 = -\frac{27}{4} = -6.75 \). Wait, but we need integer coordinates? Wait, maybe I made a mistake. Wait, \( 2^x \) when \( x \) is integer: let's check \( x = 3 \): \( 2^3 -7 = 8-7=1 \). \( x=2 \): \( 4 -7 = -3 \). \( x=1 \): \( 2 -7 = -5 \). \( x=0 \): \( 1 -7 = -6 \). \( x=-1 \): \( 0.5 -7 = -6.5 \) (not integer). \( x=3 \): \( 8 -7 =1 \). \( x=4 \): \( 16 -7 =9 \). Let's choose \( x = 0,1,2,3,4 \).

Step3: x=0

\( y = 2^0 -7 = 1 -7 = -6 \). So point (0, -6).

Step4: x=1

\( y = 2^1 -7 = 2 -7 = -5 \). Point (1, -5).

Step5: x=2

\( y = 2^2 -7 = 4 -7 = -3 \). Point (2, -3).

Step6: x=3

\( y = 2^3 -7 = 8 -7 = 1 \). Point (3, 1).

Step7: x=4

\( y = 2^4 -7 = 16 -7 = 9 \). Point (4, 9).

These points (0, -6), (1, -5), (2, -3), (3, 1), (4, 9) have integer coordinates. To graph, plot these points and draw a smooth curve through them (since it's an exponential function).

Answer:

Five points with integer coordinates for \( y = 2^x - 7 \) are (0, -6), (1, -5), (2, -3), (3, 1), (4, 9). Plot these points and draw the exponential curve.