Question

use technology to find points and then graph the function ( y = 2^x - 7 ), following the instructions below.

Explanation:

Step1: Choose x-values

Select some x - values, for example, let \(x=-3,-2,-1,0,1,2,3\).

Step2: Calculate corresponding y - values

For a given \(x\), use the function \(y = 2^{x}-7\).

• When \(x=-3\):

\(y = 2^{-3}-7=\frac{1}{2^{3}}-7=\frac{1}{8}-7=\frac{1 - 56}{8}=-\frac{55}{8}=-6.875\)

• When \(x = - 2\):

\(y=2^{-2}-7=\frac{1}{4}-7=\frac{1 - 28}{4}=-\frac{27}{4}=-6.75\)

• When \(x=-1\):

\(y=2^{-1}-7=\frac{1}{2}-7=-\frac{13}{2}=-6.5\)

• When \(x = 0\):

\(y=2^{0}-7=1 - 7=-6\)

• When \(x = 1\):

\(y=2^{1}-7=2 - 7=-5\)

• When \(x = 2\):

\(y=2^{2}-7=4 - 7=-3\)

• When \(x = 3\):

\(y=2^{3}-7=8 - 7=1\)

Step3: Plot the points

Plot the points \((-3,-6.875)\), \((-2,-6.75)\), \((-1,-6.5)\), \((0, - 6)\), \((1,-5)\), \((2,-3)\), \((3,1)\) on the coordinate plane.

Step4: Draw the graph

Connect the points with a smooth curve. The function \(y = 2^{x}-7\) is an exponential function. The parent function is \(y = 2^{x}\), which is shifted down 7 units. As \(x\to-\infty\), \(y\to - 7\) (horizontal asymptote \(y=-7\)), and as \(x\to\infty\), \(y\to\infty\).

Answer:

The graph is drawn by plotting the points \((-3,-6.875)\), \((-2,-6.75)\), \((-1,-6.5)\), \((0, - 6)\), \((1,-5)\), \((2,-3)\), \((3,1)\) and connecting them with a smooth curve, with a horizontal asymptote at \(y = - 7\).