Quadratic, cubic and reciprocal graphs：練習問題

Factorise the quadratic expression completely:
\(x^2 - 9x + 20\)

A sequence is defined by the quadratic formula for the \(n\)th term: \(u_n = an^2 + bn + c\).
The first four terms of the sequence are 5, 12, 23, and 38.
Find the expression for the \(n\)th term of this sequence.

The functions \(f\) and \(g\) are such that \(f(x) = 2x - 3\) and \(g(x) = x^2 + 1\).
Find the value of \(gf(4)\).

A sequence is defined by the term-to-term rule \(u_{n+1} = 3u_n - 4\).
Given that \(u_3 = 23\), calculate the value of \(u_1\).

Find the \(n\)-th term of the quadratic sequence:
\(5, 12, 23, 38, ...\)

A sequence is defined by the rule \(u_{n+1} = 2u_n - 3\). Given that \(u_1 = 5\), calculate the value of \(u_4\).

Write \(3x^2 + 12x - 5\) in the form \(a(x+b)^2 + c\). Hence, write down the coordinates of the turning point of the graph of \(y = 3x^2 + 12x - 5\).

Consider the function \(f(x) = x^2 + 5\) and its inverse \(f^{-1}(x)\).
Find the equation for the tangent to the curve \(y = f(x)\) at the point where the gradient of the curve is equal to the gradient of the line \(y = 6x - 1\).

Solve the simultaneous equations:
\( y = 2x + 1 \)
\( x^2 + y^2 = 13 \)

The first five terms of a quadratic sequence are: \(5, 12, 23, 38, 57\).

(a) Find an expression, in terms of \(n\), for the \(n\)th term of this sequence.
(b) The \(k\)th term of the sequence is \(530\). Find the value of \(k\).
(c) Prove that \(100\) cannot be a term in this sequence.