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What is the equation of the line tangent to the function lower f left parenthesis x right parenthesis equals 4 x squared plus 5 x at the point left parenthesis negative 2 comma 6 right parenthesis?

(1 point)
Responses

y equals negative 8 x plus 5
Image with alt text: y equals negative 8 x plus 5

y = 8x + 5
y = 8 x + 5

y equals negative 11 x plus 16
Image with alt text: y equals negative 11 x plus 16

y equals negative 11 x minus 16

1 answer

To find the equation of the tangent line to the function \( f(x) = 4x^2 + 5x \) at the point \((-2, 6)\), we need to follow these steps:

  1. Find the derivative \( f'(x) \), which gives us the slope of the tangent line.
  2. Evaluate the derivative at \( x = -2 \) to find the slope of the tangent line at that point.
  3. Use the point-slope form of the line to write the equation of the tangent line.

Step 1: Find the derivative

The derivative of \( f(x) = 4x^2 + 5x \) is calculated as follows:

\[ f'(x) = \frac{d}{dx}(4x^2) + \frac{d}{dx}(5x) = 8x + 5. \]

Step 2: Evaluate the derivative at \( x = -2 \)

Now, we plug in \( x = -2 \) into the derivative:

\[ f'(-2) = 8(-2) + 5 = -16 + 5 = -11. \]

So, the slope of the tangent line at the point \((-2, 6)\) is \(-11\).

Step 3: Use the point-slope form

The point-slope form of a line is:

\[ y - y_1 = m(x - x_1), \]

where \((x_1, y_1)\) is the point on the line and \(m\) is the slope. We have:

  • \(m = -11\)
  • \((x_1, y_1) = (-2, 6)\)

Plugging in these values gives:

\[ y - 6 = -11(x + 2). \]

Now, simplify this equation:

\[ y - 6 = -11x - 22 \] \[ y = -11x - 22 + 6 \] \[ y = -11x - 16. \]

Final Answer

The equation of the tangent line is:

\[ y = -11x - 16. \]

Therefore, the correct response is y = -11x - 16.