Free HSC Year 11 Mathematics Extension 1 Flashcard Review
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What happens to a zero of f(x) on the graph of y = 1/f(x)?
It becomes a vertical asymptote.
Where do y = f(x) and y = 1/f(x) intersect?
Where f(x) = 1 or f(x) = −1.
Range of y = sec x?
y ≤ −1 or y ≥ 1.
Asymptotes of y = cot x?
x = nπ, where sin x = 0; period π.
How do you sketch y² = f(x)?
Sketch y = √f(x) (only where f ≥ 0) and reflect it in the x-axis.
Domain of f(x) + g(x)?
The intersection of the domains of f and g.
Horizontal line test
A function is one-to-one (and has an inverse) if every horizontal line meets its graph at most once.
Domain and range of f⁻¹
Domain of f⁻¹ = range of f; range of f⁻¹ = domain of f.
Steps to find f⁻¹(x)
Write y = f(x), swap x and y, make y the subject, state the domain.
Cartesian form of x = 2at, y = at²
x² = 4ay.
How do you solve a/(x − k) ≥ b safely?
Multiply both sides by (x − k)², which is positive, then exclude x = k.
|x − a| < c as an interval
a − c < x < a + c (x lies within c of a).
What is the degree of the zero polynomial?
It has no degree, because it has no non-zero leading term.
Define a monic polynomial.
A polynomial whose leading coefficient is 1.
deg(PQ) for non-zero P and Q?
deg P + deg Q.
When can deg(P + Q) be less than max(deg P, deg Q)?
When P and Q have equal degree and their leading coefficients are opposites, so the leading terms cancel.
End behaviour of y = −2x⁵ + x?
Odd degree, negative leading coefficient: y → +∞ as x → −∞ and y → −∞ as x → +∞.
What does the graph do at a zero of multiplicity 2?
Touches the x-axis and turns back (turning point on the axis).
What does the graph do at a zero of multiplicity 3?
Passes through the axis with a horizontal point of inflexion.
State the division algorithm for polynomials.
P(x) = A(x)Q(x) + R(x), with R = 0 or deg R < deg A.