Sketching graphs
Know the shapes
A sketch is not plotted point by point, but its shape must be right.
- y = mx + c: a straight line.
- y = x² + ...: a U shape (∪). With −x² it is upside down (∩).
- y = x³ + ...: an S shape, from bottom left to top right.
- y = k/x: two separate curves that get close to the axes but never touch them.
The key points
A sketch earns its marks from the points it labels.
- y-intercept: put x = 0. For y = x² − 2x − 8 that gives y = −8
- x-intercepts: put y = 0 and solve. x² − 2x − 8 = (x + 2)(x − 4) = 0, so x = −2 or x = 4
- turning point: halfway between the roots, x = (−2 + 4) ÷ 2 = 1, then y = 1 − 2 − 8 = −9
- label all of them: (−2, 0), (4, 0), (0, −8) and the minimum (1, −9)
The usual slip: (x + 2) = 0 gives x = −2, not 2. The sign flips.
Turning points by completing the square
When the roots are not whole numbers, complete the square instead.
- x² + 6x + 4: half of 6 is 3, and (x + 3)² = x² + 6x + 9
- so x² + 6x + 4 = (x + 3)² − 5
- (x + 3)² is never negative and is 0 when x = −3
- the turning point is (−3, −5), a minimum below the x-axis, so the curve crosses the axis twice
For a cubic in brackets, each bracket gives a root. A squared bracket, such as (x − 1)², gives a point where the curve touches the axis and turns back.
Sketch and check
In the graphing calculator, type y = x^2 - 2x - 8. Check that it crosses the axes at (−2, 0), (4, 0) and (0, −8), and turns at (1, −9). Then try y = -x^2 + 4.
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