Free VCE Year 11 Mathematical Methods Flashcard Review
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What makes a relation a function?
Each input in the domain is paired with exactly one output (vertical line test).
What is the difference between co-domain and range?
The co-domain is the declared output set; the range is the set of outputs actually produced, a subset of the co-domain.
Maximal domain of f(x) = √(x − 5)?
[5, ∞), because x − 5 ≥ 0.
Does x² + y² = 9 define a function?
No. For example x = 0 gives y = 3 and y = −3, so it fails the vertical line test.
Which functions have an inverse function?
Only one-to-one functions (those passing the horizontal line test).
How are the graph, domain and range of f⁻¹ related to f?
The graph is the reflection in y = x; dom f⁻¹ = ran f and ran f⁻¹ = dom f.
Asymptotes of y = a/(x + b) + c?
Vertical x = −b and horizontal y = c.
Effect of a in y = a(x + b)ⁿ + c?
Dilation by factor |a| from the x-axis; if a < 0, also a reflection in the x-axis.
Symmetry of y = 1/x² and y = 1/x?
1/x² is even (symmetric about the y-axis); 1/x is odd (point symmetry about the origin).
What does a factor (x − k)² tell you about a polynomial graph?
The graph touches the x-axis at x = k and turns there.
End behaviour of y = −2x³ + x?
Leading term −2x³: rises on the left and falls on the right.
How do you find an average rate of change from data?
(change in y)/(change in x) between two points, stated with units.
State the exponent law for (aᵐ)ⁿ and for a⁻ⁿ.
(aᵐ)ⁿ = aᵐⁿ and a⁻ⁿ = 1/aⁿ (a ≠ 0).
Expand (a − b)(a + b).
a² − b², the difference of two squares.
How do you factorise a non-monic trinomial ax² + bx + c?
Find two numbers multiplying to ac and adding to b, split the middle term, then group.
What does the discriminant tell you?
Δ = b² − 4ac: Δ > 0 two real solutions, Δ = 0 one repeated solution, Δ < 0 no real solutions.
Complete the square for x² + bx + c.
(x + b/2)² − (b/2)² + c.
Remainder theorem
Dividing p(x) by (x − a) leaves remainder p(a).
Graph behaviour at a repeated factor (x − a)²
The graph touches the x-axis at (a, 0) and turns back.
Rational root theorem
A rational root p/q (lowest terms) of an integer polynomial has p dividing the constant term and q dividing the leading coefficient.