{Use of Tech} Equations of tangent lines
Find an equation of the line tangent to the given curve at a.
y = −3x² + 2; a=1

Verified step by step guidance

Step 1: Identify the function and the point of tangency. The function given is \( y = -3x^2 + 2 \) and the point of tangency is at \( x = a = 1 \).

Step 2: Find the derivative of the function to determine the slope of the tangent line. The derivative of \( y = -3x^2 + 2 \) with respect to \( x \) is \( y' = \frac{d}{dx}(-3x^2 + 2) = -6x \).

Step 3: Evaluate the derivative at \( x = 1 \) to find the slope of the tangent line at that point. Substitute \( x = 1 \) into \( y' = -6x \) to get the slope \( m = -6(1) = -6 \).

Step 4: Find the y-coordinate of the point of tangency by substituting \( x = 1 \) into the original function. \( y = -3(1)^2 + 2 = -3 + 2 = -1 \). So, the point of tangency is \( (1, -1) \).

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 and \( (x_1, y_1) \) is the point of tangency. Substitute \( m = -6 \), \( x_1 = 1 \), and \( y_1 = -1 \) into the equation to get \( y - (-1) = -6(x - 1) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which can be found using the derivative of the function.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. For the curve given, the derivative will provide the slope of the tangent line at the specified point.

Point-Slope Form

The point-slope form of a linear equation is used to write the equation of a line when the slope and a point on the line are known. It is expressed as y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful for finding the equation of the tangent line once the slope has been determined.

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