Level 7: Quadratics and functions

Algebra · level 7 of 8 · rough age 15-16 (a guide for parents; anyone can start anywhere)

In level 6 you multiplied and factored polynomials and solved pairs of straight-line equations. In this level the straight line bends. You meet the parabola, the curve of every thrown ball, learn its three forms, and solve any quadratic equation, finishing with a formula you derive yourself. Then the idea of a function takes centre stage: what it is, how to move and stretch its graph, how to undo it, and the first function whose unknown lives in the exponent.

Lesson	You will be able to
Quadratic relations and the parabola	Read the vertex, axis of symmetry and zeros from the standard, factored and vertex forms, and explain why the axis lies halfway between the zeros
Completing the square	Turn \(ax^2 + bx + c\) into vertex form with a square picture, and find the greatest height of a ball or the largest area of a fence
Solving quadratic equations	Solve by factoring, by completing the square and by the quadratic formula, derive the formula, and count the roots with the discriminant
Functions	Use \(f(x)\) notation, find domain and range, and decide with mapping diagrams and the vertical line test whether a relation is a function
Transformations of functions	Sketch \(y = af(k(x - d)) + c\) from \(x^2\), \(\sqrt{x}\), \(1/x\) and \(\lvert x \rvert\), in the right order, with a mapping rule
Inverse functions	Find an inverse by swapping \(x\) and \(y\), draw it as a reflection in \(y = x\), and restrict a domain so that an inverse exists
Exponential functions	Evaluate rational exponents like \(8^{2/3}\), graph \(y = a \cdot b^x\) with its asymptote, model growth and decay, and solve \(2^x = 32\)

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