Graphs of functions and sketching curves

• Construct tables of values and draw graphs of y = axⁿ (sums of up to three such terms, n = −2, −1, −½, 0, ½, 1, 2, 3) and y = ab^x + c.
• Solve equations graphically, including finding roots and the intersection of a line and a curve.
• Draw and interpret exponential growth and decay graphs.
• Estimate the gradient of a curve by drawing a tangent.
• Recognise and sketch linear, quadratic, cubic, reciprocal and exponential graphs, with roots, turning points, symmetry and asymptotes.

Substitute carefully, especially negatives and fractions

Work out y for each x in the table, using brackets for negative values. Plot the points and join them with a smooth curve (not straight segments). For a curve with x in the denominator, there is a break where the denominator is zero.

Read the x-coordinates where graphs meet

The roots of f(x) = 0 are the x-coordinates where y = f(x) crosses the x-axis. To solve a different equation using a graph you already have, rearrange it so one side is the plotted curve; the other side is a line to draw. The solutions are the x-coordinates of the intersections.

Growth, decay and tangents

y = ab^x + c with b > 1 shows exponential growth; with 0 < b < 1 it shows decay. The curve approaches the line y = c but never reaches it. Read values from the graph to answer questions such as “when does the value fall below …?”. To estimate the gradient at a point, draw a tangent and find its gradient.

Know each family on sight

asymptote ^{−2 1/2}