The table of values below is for the curve \(y = 2^x - 3\). \(x\)\(-1\)\(-0.5\)\(0\)\(0.5\)\(1\)\(1.5\)\(2\)\(2.5\)\(3\) \(y\)\(-2.5\)\(-2.3\)\(p\)\(-1.6\)\(-1\)\(-0.2\)\(1\)\(2.7\)\(5\) Find the value of \(p\). Using a scale of \(4\) cm to represent \(1\) unit on the horizontal \(x\)-axis and \(2\) cm to represent \(1\) unit on the vertical \(y\)-axis, draw the graph of \(y = 2^x - 3\) for \(-1 \le x \le 3\). Use your graph to find the value of \(x\) when \(y = 3.5\), the gradient of the curve at \(x = 2\). On the same graph, draw the line \(y = -x\). Hence, find the solution to the equation \(2^x - 3 = -x\).

Exponential and Logarithmic Functions - Sketching Exponential Functions and Solving Equations Graphically

Secondary 3

The table of values below is for the curve \(y = 2^x - 3\).

\(x\)	\(-1\)	\(-0.5\)	\(0\)	\(0.5\)	\(1\)	\(1.5\)	\(2\)	\(2.5\)	\(3\)
\(y\)	\(-2.5\)	\(-2.3\)	\(p\)	\(-1.6\)	\(-1\)	\(-0.2\)	\(1\)	\(2.7\)	\(5\)

Find the value of \(p\).
Using a scale of \(4\) cm to represent \(1\) unit on the horizontal \(x\)-axis and \(2\) cm to represent \(1\) unit on the vertical \(y\)-axis, draw the graph of \(y = 2^x - 3\) for \(-1 \le x \le 3\).
Use your graph to find
1. the value of \(x\) when \(y = 3.5\),
2. the gradient of the curve at \(x = 2\).
1. On the same graph, draw the line \(y = -x\).
2. Hence, find the solution to the equation \(2^x - 3 = -x\).

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Answer:(a) \(-2\). (c)(i) \(2.70\); (ii) about \(2.77\). (d)(ii) \(1\).

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