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A cyclist follows the curve y = x² - 4x + 7 on a map, with x and y in kilometres…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A cyclist follows the curve $y = x^2 - 4x + 7$ on a map, with $x$ and $y$ in kilometres. At the point where $x = 3$ , find the gradient of the tangent and the $y$ -intercept of the tangent line. Then find the gradient and the $y$ -intercept of the normal line at the same point.

Need a reminder? The key ideaShow

The tangent to $y = f(x)$ at $(x_0, y_0)$ is $y - y_0 = f'(x_0)(x - x_0)$ . The normal has gradient $-\dfrac{1}{f'(x_0)}$ and passes through the same point. A horizontal tangent has a vertical normal.

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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