In the upper half of this applet, the graph of \(y=f(x)\) is drawn, where \(f(x)=ax^2+bx+c\). You can change the values of the coefficients \(a\), \(b\) and \(c\) by moving the sliders or typing in values.

The points \((x,f(x))\) are shown in green for \(x=-4\), \(-3\), …, \(4\) (though some of these may be beyond the visible part of the graph).

Part of the tangent to the graph of \(y=f(x)\) is drawn in orange at each of these points.

The gradients of these tangents have been calculated; they are shown in the spreadsheet on the right hand side, and are also plotted on the graph in the lower half of the applet.

The big question: Can you find a relationship between the equation of the quadratic and its gradient at \(x\)?

For each of the following questions, describe what you think will happen before you try it out on the interactivity. Then reflect on your ideas: did the actual behaviour surprise you? And can you explain the observed behaviour? (You might want to start with very simple values for \(a\), \(b\) and \(c\).)

What happens to the gradients as you change \(c\)?

What happens to the gradients as you change \(a\)?

What happens to the gradients as you change \(b\)?

Given what you have discovered, can you use it to predict what the gradients will be for some or all of the following quadratics? Test your predictions!

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

Now predict the gradients for some other quadratics and test your results until you are confident that you have found a way of predicting the gradient function for any quadratic.