A graphing celculator is recommended. (a) The curve with ecuation \( y^{2}=x^{3}+3 x^{2} \) is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point ( 1,2 ) (t) At what points does this curve have horizontal tangents? (Order your answers from smallest to largest \( x \), then from smallest to largest \( y \) ) \( \qquad(x, y)= \) ( \( x, y)= \) (b) by graphing the curve and the tangent lines on a common screen.

It seems that the system does not currently support plotting the graphs directly. However, I can guide you on how to graph the curve and the tangent line using graphing software or a graphing calculator. ### Instructions for Graphing 1. **Graph the Curve**: - Use the equation \( y^2 = x^3 + 3x^2 \). - You can rearrange it to \( y = \pm \sqrt{x^3 + 3x^2} \) to plot both the positive and negative branches of the curve. 2. **Graph the Tangent Line**: - Use the equation \( y = \frac{9}{4}x - \frac{1}{4} \). - This is a straight line with a slope of \( \frac{9}{4} \) and a y-intercept of \( -\frac{1}{4} \). 3. **Set the Viewing Window**: - For both graphs, set the x-axis from approximately -4 to 2 and the y-axis from -3 to 3 to capture the relevant features of the curve and the tangent line. ### Points to Mark - Mark the point of tangency (1, 2) on the graph. - Also, mark the points where the curve has horizontal tangents: \( (-2, -2) \), \( (-2, 2) \), and \( (0, 0) \). ### Conclusion By following these instructions, you should be able to visualize the Tschirnhausen cubic and its tangent line effectively. If you have access to graphing software like Desmos, GeoGebra, or a graphing calculator, you can input the equations directly to see the graphs.