Question

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

Answer 1

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**.