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Think of the types of graphs you can obtain by a combination of stretches, reflections and translations of the graph $y=\sin x$. In this resource we refer to any of these graphs as a "sine graph".
Can you find ...
(a) ... a sine graph which touches the lines $y=3$ and $y=1?$
(b) ... a cosine graph which crosses the $x$-axis at $x=1$ and $x=-1$?
(c) ... a tangent graph which passes through the point $\big(\dfrac{\pi}{3},0\big)$ and for which the line $x=\dfrac{\pi}{2}$ is an asymptote?
You can use the free Desmos graphing calculator to help you find suitable graphs, but try to sketch some graphs first.
This comes in two parts, with the first being less fiendish than the second. It’s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.
You're invited to decide whether statements about the number of solutions of a quadratic equation are always, sometimes or never true.
This will encourage you to think about whether all quadratics can be factorised and to develop a better understanding of the effect that changing the coefficients has on the factorised form.