Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = 2x² / (3x - 1); a = 1

Verified step by step guidance

1. Step 1: Identify the function and the point of tangency. The function given is \( y = \frac{2x^2}{3x - 1} \) and the point of tangency is at \( x = 1 \).
2. Step 2: Find the derivative of the function to determine the slope of the tangent line. Use the quotient rule: if \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \). Here, \( u = 2x^2 \) and \( v = 3x - 1 \).
3. Step 3: Calculate \( u' \) and \( v' \). For \( u = 2x^2 \), \( u' = 4x \). For \( v = 3x - 1 \), \( v' = 3 \). Substitute these into the quotient rule formula to find \( y' \).
4. Step 4: Evaluate the derivative at \( x = 1 \) to find the slope of the tangent line. Substitute \( x = 1 \) into the derivative \( y' \) to get the slope \( m \).
5. Step 5: Use the point-slope form of a line to write the equation of the tangent line. The point-slope form is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope found in Step 4, and \( (x_1, y_1) \) is the point \( (1, y(1)) \). Calculate \( y(1) \) using the original function.