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.

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between variables. In this case, the function y = 2x² / (3x - 1) represents a rational function, which can exhibit various behaviors such as asymptotes and intercepts. Understanding how to graph this function is essential for analyzing its shape and identifying key features.

Tangent Lines

A tangent line to a curve at a given point represents the instantaneous rate of change of the function at that point. It is defined by the derivative of the function evaluated at that point. For the function y = 2x² / (3x - 1), finding the tangent line at x = 1 requires calculating the derivative and using the point-slope form of a line.

Derivatives

The derivative of a function measures how the function's output changes as its input changes, providing a way to determine slopes of tangent lines. For the function y = 2x² / (3x - 1), applying the quotient rule will yield the derivative, which is crucial for finding the slope of the tangent line at the specified point. Understanding derivatives is fundamental in calculus for analyzing function behavior.