The table shows values for \( y = 2^x – 3 \).
(a) Complete the table.
(b) Draw the graph of \( y = 2^x – 3 \) for \( -2 \leq x \leq 2.5 \).
(c) Use the graph to solve \( 2^x – 3 = 2 \).
(d) By drawing a suitable straight line, solve \( 2^x – x – 1.5 = 0 \).
▶️ Answer/Explanation
(a) -2.5, -2, -1 (table values)
Calculate y when x=-1: 2⁻¹-3=0.5-3=-2.5; x=0: 2⁰-3=1-3=-2; x=1: 2¹-3=2-3=-1.
(b) Correct graph
Plot points (-2,-2.75), (-1,-2.5), (0,-2), (1,-1), (2,1) and (2.5,1.828) from table. Connect with smooth exponential curve.
(c) 2.3 to 2.4
Find x-value where graph crosses y=2 line. Solution is approximately x≈2.32.
(d) -1 and 1.55 to 1.7
Rearrange to 2^x-3=x-1.5. Draw line y=x-1.5. Solutions are x-coordinates where this line intersects the original graph.
The table shows some values for \(y=x^{2}-\frac{1}{3x},x\neq 0\)
The y-values are rounded to 1 decimal place.
(a) Complete the table.
(b) On the grid, draw the graph of \( y=x^{2}-\frac{1}{3x} \) for \(-2\leqslant x\leqslant -0.1\). The graph of \( y=x^{2}-\frac{1}{3x} \) for x > 0 has been drawn for you.
(c) By drawing a suitable line on the grid, solve the equation \(y=x^{2}-\frac{1}{3x} +1=0\)
▶️ Answer/Explanation
(a) 1[.0], 0.9
For x=1: y=1²-1/(3×1)=0.666…≈0.7. For x=0.5: y=0.5²-1/(3×0.5)≈0.25-0.666≈-0.4.
(b) correct curve
Plot the points from the table and connect them smoothly, mirroring the given positive-x curve.
(c) ruled line at y = −1, 0.3 to 0.32
Rewrite equation as x²-1/(3x)=-1. Draw y=-1 line, find intersection with curve for solution.
