Question

Group Activity Using graphing calculators, have each person in your group do the following: (a) pick two numbers \(a\) and \(b\) between 1 and \(10 ;\) (b) graph the function \(y=(x-a)(x+b)\) ; (c) graph the derivative of your function (it will be a line with slope 2\()\) (d) find the \(y\) -intercept of your derivative a simple way to predict the \(y\) -intercept, given the values of \(a\) and \(b\) . Test your result.

Short answer

Calculated derivative is \(y’=2x-(b+a)\) and its y-intercept is \(- (b+a)\).

Step-by-step solution

Choose Two Numbers

Choose two numbers \(a\) and \(b\) between 1 and 10.

Graph The Function

After having \(a\) and \(b\), plot the function \(y=(x-a)(x+b)\) on the graph. Use several points to have an idea of the function shape.

Find the Derivative

Find the derivative of the function. The derivative of \(y=(x-a)(x+b)\) with respect to \(x\) is \(y’=2x-(b+a)\). This equation can be calculated using the power rule of differentiation.

Graph the Derivative

Graph the derivative on the same plot. As mentioned, it will be a straight line since its equation is a linear.

Find the Y-Intercept

Find the y-intercept of your derivative. As per the equation of a line \(y = mx + c\), the y-intercept is \(c\), which happens when \(x=0\). So, substitute \(x = 0\) into the derivative function: \(y’=(2*0)-(b+a) = - (b+a)\). This value is the y-intercept.

Test your Result

Check the previously found y -intercept on the derivative graph. It should cross the y-axis at this point.