Germany Jahrgangsstufe 8 Mathematics
Chapters: 7
1. Functions and terms
Functions and Terms
- Functions: Input, Rule, Output – A function is a rule that gives exactly one output for each input. We write it as f(x). The output f(a) is called the image of a; an input that gives a certain output is a preimage. A function can be shown as words, a table, a formula or a graph. Its graph can go up (increasing), go down (decreasing) and have a highest point (maximum) or lowest point (minimum).
2. Linear functions
Linear Functions
- Linear Functions – A linear function adds the same amount to y every time x goes up by 1. Its rule is y = mx + c (also written f(x) = mx + b). m is the slope: rise ÷ run, the change in y for each 1 step in x. c is the y-intercept: the value of y when x = 0. Its graph is always a straight line.
3. Basic rational functions
Simple Rational Functions
- Rational Functions: Graphs, Asymptotes and Sketching – A rational function is one polynomial divided by another, like y = (2x + 1)/(x − 1). It is not defined where the bottom is zero. Near those x-values the graph shoots up or down along a vertical asymptote. Far out, it settles near a horizontal (or slant) asymptote. Find domain, asymptotes, intercepts and holes, then sketch.
4. Algebraic fractions and fractional equations
Algebraic Fractions and Fraction Equations
Coming soon
5. Laplace experiments (equally likely outcomes)
Laplace Experiments
- Introduction to Probability: Scale, Experiments, Sample Spaces and Trees – Probability is a number from 0 to 1 that tells how likely something is. 0 means it can never happen, 1 means it will surely happen. We can find it by doing an experiment many times (empirical probability), or by listing every possible result (the sample space) and counting the ones we want. Tree diagrams and tables help us list results when two things happen together.
6. Systems of linear equations
Systems of Linear Equations
- Pair of Linear Equations in Two Variables – Two equations like a₁x + b₁y = c₁ and a₂x + b₂y = c₂ each make a straight line. The answer that fits both is the point where the lines meet. Lines that cross give one answer, parallel lines give none, and lines that lie on top of each other give endless answers. We can find the answer by drawing (graph), by substitution or by elimination.
7. Circle and cylinder
Circle and Cylinder
- Surface Area and Volume of Cuboid, Cylinder, Cone, Pyramid and Sphere – Surface area is the total area of the outside skin of a solid, like the paper needed to wrap it. Volume is the space inside, like the water it can hold. A cuboid's volume is the number of 1 cm cubes that fit in it. A cone holds one third of a cylinder with the same base and height, and a pyramid holds one third of the matching prism.