This unit starts with the idea that makes the rest possible: a function gives every input exactly one output. Students test relations with the vertical line test, state a domain and range, and use the notation f(x).
The middle of the unit is the quadratic. Factorizing reverses expanding, a quadratic function has three forms that each show a different feature of the same parabola, and the null factor law turns a product equal to zero into two simple equations. When factorizing fails, completing the square and the quadratic formula still work.
The last lesson names the part of the formula under the root, the discriminant, and uses it to count real solutions and to choose a method. 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; Lesson 2 is done by hand.
The lessons, in order
- Lesson 1 · Functions, domain and range Relations as ordered pairs, graphs and equations, the vertical line test, domain and range, f(x) notation, evaluating f at a number and finding the input for a given output. Slide a vertical line across five relations and count how many times it meets each one. GDC optional
- Lesson 2 · Factorizing quadratics Factorizing x² + bx + c by a product and a sum, the difference of two squares, taking out a common factor, and factorizing ax² + bx + c by splitting the middle term. Set b and c, see every pair of whole numbers with product c, and see which pair adds to b; cut a corner off a square and rearrange it. No calculator
- Lesson 3 · The three forms of a quadratic The three forms of a quadratic function, the vertex, the axis of symmetry x = −b ÷ (2a), zeros, the y-intercept, and whether the parabola opens up or down. Move a, h and k and watch one parabola written three ways, with its vertex, zeros and y-intercept marked. GDC optional
- Lesson 4 · The null factor law and solving by factorizing The null factor law, rearranging a quadratic equation to equal zero, solving by factorizing, the solution x = 0, repeated solutions, and rejecting a solution that does not fit the context. Move two brackets' zeros and the input x, and see that the product is zero exactly when a bracket is. GDC optional
- Lesson 5 · Completing the square and the quadratic formula Completing the square for x² + bx + c, solving by completing the square, the quadratic formula, exact answers with a surd, and rounding to decimal places. Build x² + 2mx from a square and two strips, see the missing corner, and watch the solutions follow. GDC optional
- Lesson 6 · The discriminant and choosing a method The discriminant Δ = b² − 4ac, the number of real solutions, when a quadratic factorizes, finding a parameter for a given number of solutions, and choosing the best method. Move a, b and c and watch the discriminant decide whether the parabola crosses, touches or misses the x-axis. GDC optional
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.