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Graphing functions involves plotting points on a coordinate plane to visually represent a mathematical equation. This visualization helps in understanding the behavior of the function. For the given function \(y=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}+2 x\), using a graphing utility is a great way to identify where horizontal tangent lines might occur.When graphing, pay attention to:

• The shape of the graph: This function, being a cubic polynomial, will likely have two turning points.
• Where the curve appears to flatten out, or plateau—these areas are candidates for horizontal tangents.

Once you have a graph, it becomes easier to estimate the x-values where these flat, horizontal tangents occur. Graphing functions allows students to draw a link between algebraic expressions and geometric representations.

Differentiation is a core concept in calculus used to find the rate at which one quantity changes with respect to another. It is the process of finding the derivative of a function. The derivative represents the slope of the tangent line to the curve at any given point.In the context of our function:- The original equation is \(y=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}+2 x\).- Its derivative, \(y' = x^2 - 3x + 2\), gives us another equation representing the slope of the curve at any point.- For horizontal tangent lines, which have a slope of zero, the derivative \(y'\) is set to zero, resulting in the equation \(x^2 - 3x + 2 = 0\). Solving this derivative equation reveals where the function has horizontal tangents, a legendary calculus problem helping students join the dots between graphs and derivatives.

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). Solving them is fundamental to understanding various algebraic problems.For our differentiated equation \(x^2 - 3x + 2 = 0\):

• Recognize \(a = 1\), \(b = -3\), \(c = 2\).
• Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) to find the roots of the equation.

Plugging in the values:- The discriminant \(b^2 - 4ac = 9 - 8 = 1\) is calculated, leading to a positive value indicating two real roots.- The roots are \(x = \frac{3 \pm 1}{2}\), giving solutions \(x = 2\) and \(x = 1\).These solutions provide the x-values where the original function has horizontal tangents. Thus, quadratic equations bridge the process of differentiation to specific problem-solving tasks in graph analysis.