This unit starts with what a function accepts and what it can give: domain, range and notation with an expression as the input. It then treats quadratics at depth. Factorizing is extended to a not equal to 1 and given a proof, the three forms are joined by completing the square, and a quadratic is fitted to data from points, a vertex or zeros.
Solving quadratics is the next step: the formula is derived, and the discriminant counts solutions and sets parameters. The last two lessons move to any function. Changing the rule moves, flips and stretches its graph, and combining transformations raises the question of order and of when two transformations are the same.
A graphic display calculator (GDC) is optional all the way through. Lessons 1, 3, 4, 5 and 6 have a closed “With a GDC” section to open if you have one; everything else is done by hand.
The lessons, in order
- Lesson 1 · Functions: notation, domain and range Function notation with an expression as the input, the natural domain (denominators and square roots), and the range of a function, including a quadratic on a restricted domain. Choose a function and slide the input to see which values are accepted, which are refused, and where the outputs stop. GDC optional
- Lesson 2 · Factorizing every type of quadratic Factorizing ax² + bx + c with a not equal to 1, the common factor first, the difference of two squares and perfect squares, why the method works, and spotting a quadratic with no whole-number brackets. Set a, b and c and see the pair that splits b, the brackets, and the zeros, or why no whole-number brackets exist. No calculator
- Lesson 3 · Three forms and completing the square Standard, factorized and vertex form; completing the square including a not equal to 1; the vertex (−b ÷ (2a), c − b² ÷ (4a)); and the least or greatest value of a quadratic model. Set a, b and c in standard form and see the vertex form, the vertex, the zeros and the y-intercept at once. GDC optional
- Lesson 4 · Fitting a quadratic model Finding y = ax² + bx + c from three points, from the vertex and a point, or from the zeros and a point; choosing the form; when three points lie on a line. Set the heights of a curve at three places and see the one quadratic through them, or the line they make. GDC optional
- Lesson 5 · Solving quadratics and the discriminant Solving by factorizing, completing the square and the quadratic formula; deriving the formula; the discriminant and the number of real solutions; choosing the method. Move b and watch the parabola cross, touch or miss the axis, beside the value of the discriminant. GDC optional
- Lesson 6 · Translating, reflecting and dilating a graph Translations f(x − h) + k, reflections −f(x) and f(−x), and dilations af(x) and f(bx) of any function, and the position of a tracked point. Move, flip and stretch a zigzag graph with sliders and buttons, and follow one point to its new position. GDC optional
- Lesson 7 · Combining transformations, and when two agree Combinations of translations, reflections and dilations; the order of transformations; writing the rule from a description; f(bx − h); and f(bx) = bⁿf(x) for power functions. Stretch and shift a graph in either order and see when the results agree, then see what f(bx − h) really shifts by. No calculator
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.