### Calculus of Powers

Building blocks

• 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.