Advanced Algebra
Equations and Curves
Move between equations and the curves they draw: hyperbolas and how they shift and stretch, rational functions and their asymptotes, square roots and other roots, where graphs meet, circles and ellipses, and regions cut out by inequalities.
THE SYLLABUS
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Points on a Curve
A point (x, y) is on the graph of an equation when it makes the equation true, and such a point is a solution. Putting in a value of x finds the matching values of y, sometimes two of them, and all of the solutions together form the curve.
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Identifying Equations
When a curve has to pass through a point, putting that point into the equation turns it into an equation for the missing number. One point can fix one missing number, and two points can fix two.
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Equations for the Same Curve
Equivalent equations have exactly the same solutions, so they draw identical graphs. Adding the same amount to both sides, or multiplying both sides by the same non-zero number, turns an equation into an equivalent one that can look very different.
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Bounded Curves
A bounded curve fits inside a box, and its highest, lowest, leftmost and rightmost points can be read from the graph or reasoned from the equation: a term that can never be negative caps how big the other one can get.
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Identifying Curves
An equation gives clues to its graph before anything is drawn: whether it is straight or curved, which quadrants it passes through, whether it stays above or below an axis, and whether it closes up around a centre.
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The Graph of a Function
A function gives exactly one output for each input, so no vertical line can cross its graph more than once: the vertical line test. A curve that fails it can still be the graph of a function on an interval where it passes.
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Level Check
Fifteen questions from Level 1: points on curves, missing numbers found from points, equivalent equations, a curve read for its crossing, and five two-line systems built from their solution.
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III
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VI
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