Question

7b hence, plot $y = 2^x - 3$ on the same graph as $y = 2^x$.

Explanation:

Step1: Identify parent function points

For \(y=2^x\), key points: when \(x=0\), \(y=1\); \(x=1\), \(y=2\); \(x=-1\), \(y=0.5\).

Step2: Apply vertical shift

Shift each point of \(y=2^x\) down by 3 units:

• \(x=0\): \(1-3=-2\) (point \((0,-2)\))
• \(x=1\): \(2-3=-1\) (point \((1,-1)\))
• \(x=-1\): \(0.5-3=-2.5\) (point \((-1,-2.5)\))
• As \(x\to-\infty\), \(2^x\to0\), so \(y\to-3\) (horizontal asymptote \(y=-3\)).

Step3: Plot the shifted graph

Draw the curve through the shifted points, approaching \(y=-3\) as \(x\to-\infty\), on the same axes as \(y=2^x\).

Answer:

Plot \(y=2^x - 3\) by shifting \(y=2^x\) down 3 units, with asymptote \(y=-3\) and points like \((0,-2)\), \((1,-1)\).