The diagram shows the graph of \(y=|\mathrm{f}(x)|\), where \(\mathrm{f}\) is a cubic polynomial.

Find expressions for the two possible functions \(\mathrm{f}(x)\). Write each expression in fully factorised form.

(a)

The diagram shows the graph of \(y=(2 x+a)(x+b)(x+c)\) where \(a, b\) and \(c\) are integers. Find values for \(a, b\) and \(c\).

(b) Use the graph to find the values of \(x\) for which \(y \geqslant 2\).

(a) Find the \(x\)-coordinates of the stationary points on the curve \(y=\frac12(3-2x)(x+2)^2\).

(b) On the axes, sketch the graph of \(y=\frac12(3-2x)(x+2)^2\), stating the intercepts with the coordinate axes.

(c) Find the values of \(k\) for which the equation \(\frac12(3-2x)(x+2)^2=k\) has three real and distinct roots.

The diagram shows the graph of \(y=|\mathrm{f}(x)|\), where \(\mathrm{f}(x)\) is a cubic polynomial. Find the two possible expressions for \(\mathrm{f}(x)\) in terms of linear factors with integer coefficients.

It is given that \(y=\mathrm{f}(x)\), where \(\mathrm{f}(x)=(2 x-5)(x-1)^{2}\). (a) Find the coordinates of the stationary points on the curve \(y=\mathrm{f}(x)\).

(b) On the axes, sketch the graph of \(y=\mathrm{f}(x)\), stating the intercepts with the axes.

(c) Hence find the values of \(k\) for which \(\mathrm{f}(x)=k\) has exactly one solution.

(a) Find the coordinates of the stationary points on the curve \(y=(2 x+1)^{2}(x-3)\). (b) On the axes, sketch the graph of \(y=(2 x+1)^{2}(x-3)\), stating the intercepts with the axes. (c) Write down the values of \(k\) for which the equation \((2 x+1)^{2}(x-3)=k\) has exactly one solution.

(a) On the axes, sketch the graph of \(y=(2 x-5)(x+3)(1-x)\), stating the intercepts with the coordinate axes.

(b) Hence (i) solve the inequality \((2 x-5)(x+3)(1-x) \leqslant 0\)

(ii) on the axes below, sketch the graph of \(y=|(2 x-5)(x+3)(1-x)|\).

A curve has equation

\(y=(5-x)(x+2)^2.\)

(a) Find the \(x\)-coordinates of the stationary points on the curve.

(b) Sketch the graph of \(y=(5-x)(x+2)^2\), stating the coordinates of the points where the curve meets the axes.

(c) Find the values of \(k\) for which the equation

\(k=(5-x)(x+2)^2\)

has one distinct root only.

(a) The diagram shows the graph of \(y=\lvert f(x)\rvert\), where \(f(x)\) is a cubic polynomial. Find, in factorised form, the possible expressions for \(f(x)\).

(b) Solve the inequality

\(\lvert5x-2\rvert\leq\lvert4x+1\rvert.\)

The diagram shows the graph of \(y=h(x)\), where

\(h(x)=(x+a)^2(b+cx)\)

and \(a\), \(b\) and \(c\) are integers. The curve meets the \(x\)-axis at the points \((-2,0)\) and \((1.5,0)\), and the \(y\)-axis at the point \((0,12)\).

(a) Find the values of \(a\), \(b\) and \(c\).

(b) Use the graph to solve the inequality \(h(x)\leq9\).

(a) The diagram shows the graph of \(y=|f(x)|\), where \(f(x)\) is a cubic. Find the possible expressions for \(f(x)\).

(b)(i) Sketch \(y=|2x+1|\) and \(y=|4(x-1)|\), stating the intercepts.

(b)(ii) Find the exact solutions of \(|2x+1|=|4(x-1)|\).

(a) Show that the equation of the curve

\(y=(x^2-4)(x-2)\)

can be written as

\(y=x^3+ax^2+bx+8,\)

where \(a\) and \(b\) are integers. Hence find the exact coordinates of the stationary points on the curve.

(b) On the axes, sketch the graph of

\(y=\left|(x^2-4)(x-2)\right|,\)

stating the intercepts with the coordinate axes.

(c) Find the possible values of the constant \(k\) for which

\(\left|(x^2-4)(x-2)\right|=k\)

has exactly \(4\) different solutions.

On the axes, sketch the graph of

\(y=3(x-3)(x-1)(x+2),\)

stating the intercepts with the coordinate axes.

(a) Find the coordinates of the stationary points on the curve \(y=(2x+1)(x-3)^2\). Give your answers in exact form.

(b) On the axes below, sketch the graph of \(y=\left|(2x+1)(x-3)^2\right|\), stating the coordinates of the points where the curve meets the axes.

(c) Hence write down the value of the constant \(k\) such that \(\left|(2x+1)(x-3)^2\right|=k\) has exactly 3 distinct solutions.

The diagram shows the graph of the cubic function \(y=\mathrm f(x)\). The intercepts of the curve with the axes are all integers.

(a) Find the set of values of \(x\) for which \(\mathrm f(x)\lt 0\).

(b) Find an expression for \(\mathrm f(x)\).

On the axes below, sketch the graph of

\(y=-\frac14(2x+1)(x-3)(x+4),\)

stating the intercepts with the coordinate axes.

Sketch the graph of

\(y=\left|(x-2)(x+1)(x+2)\right|.\)

Show the coordinates of the points where the graph meets the axes.

(a) Sketch the graph of

\(y=-(x+2)(x-1)(x-6),\)

showing the coordinates of the points where the graph meets the coordinate axes.

(b) Hence solve

\(-(x+2)(x-1)(x-6)\leq0.\)