The table gives some values of $x$ and the corresponding values of $y$, where \(y = 30 - 18

Question

The table gives some values of $x$ and the corresponding values of $y$, where $y = 30 - 18x + x^3$. $x$$-4$$-3$$-2$$-1$$0$$1$$2$$3$$4$ $y$$38$$57$$58$$47$$30$$p$$2$$3$$22$ Calculate the value of $p$. Using a horizontal scale of $2$ cm to represent $1$ unit and a vertical scale of $2$ cm to represent $10$ units, draw the graph of $y = 30 - 18x + x^3$. Use your graph to find the largest value of $30 - 18x + x^3$, the possible solutions of $30 - 18x + x^3 = 10$. By drawing a tangent, find the gradient of the curve when $x = 2$. On the same axes, draw the graph of the straight line $y = -8x + 27$. Find the equation, in the form of $x^3 + ax + b = 0$, which is satisfied by the values of $x$ at the points where the two graphs meet. For the graph readings in (b) and (c), use the table interval $-4\le x\le4$. The cubic has no largest value over all real inputs.

Accepted Answer

(a) $13$. (c)(i) About $59.4$ on $[-4,4]$; (ii) $1.21,3.51$. (d) About $-6$. (f) $x^3-10x+3=0$.

Functions and Graphs - Estimating Gradients from Tangents

Functions and Graphs - Estimating Gradients from Tangents

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