Maths 1.1Lesson outlineLesson 1 of 3 in Module 1

Functions and Their Graphs

Read a function as a rule from input to output, and recognise the shapes ML keeps using.

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Learn9 parts Key terms17 terms Quiz5 questions PracticeProblems A–C DoneResult and what’s next

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In this lesson9 parts

01The error curve as a function
02Input, output, graph and domain
03Linear, quadratic, exponential, sigmoid and logarithm
04Rate of change
05Shifting and stretching a graph
06Checkpoint: functions, shapes and transformations
07Age against price on the used-car lot
08Shapes and the lot in NumPy
09Recap, challenge, and next: exponents and logarithms

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Key terms

The words this lesson introduces, each in one line. The module’s glossary collects them all.

function: a rule that gives exactly one output for each input | f(x) = 2x + 1, f(2) = 5
input**, **output: the value a function takes in, and the one it gives back | 2 in, 5 out
graph: every (input, output) pair of a function, drawn as points | the line through (−2, −3) … (2, 5)
domain: the inputs a function accepts | every real number for 2x + 1; positive numbers for log₂ x
linear: a straight-line graph, ax + b | 2x + 1
quadratic: highest power two; its graph is a parabola | x²
exponential: the input is the exponent, bˣ; each equal step multiplies the output by the same factor | 2ˣ doubles at every step
sigmoid: 1 / (1 + e⁻ˣ), an S-shaped curve between zero and one | 0.5 at x = 0
logarithm: log₂ x is the power two must be raised to to give x | log₂ 4 = 2
rate of change (slope): change in output divided by change in input | 2 everywhere for 2x + 1
stretch: multiplying the output by a | 2x²
shift: adding k, or replacing x with x − h | (x − 3)² + 1
vertex: the lowest or highest point of a parabola | (3, 1) for 2(x − 3)² + 1
linear model: a prediction that is a straight-line function of its input | price = 21.425 − 1.228 · age
residual: actual value minus predicted value | the middle third's mean, −1.34
common mistake: f(x − h): it moves the graph right by h; find the input that makes the bracket zero | (x − 3)² moves right
common mistake: the line through the middle: a line through the middle of the points can still be the wrong shape; read the residual pattern | + − + by age third

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