Use the four-step process to find the slope of the tangent line to the graph of the function at the given point and determine an equation of the tangent line.
f(x) = -2x2-2x+1 (-1, -2)
f'(x) = -4x-2
f'(-1) = 2
point-slope form:
y+2 = 2(x+1)
To find the slope of the tangent line to the graph of the function at the given point and determine an equation of the tangent line, we can use the four-step process:
Step 1: Find the derivative of the function.
Step 2: Evaluate the derivative at the given x-value.
Step 3: Calculate the slope of the tangent line using the derivative.
Step 4: Use the point-slope form of a line to find the equation of the tangent line.
Let's go through each step in detail:
Step 1: Find the derivative of the function.
The given function is f(x) = -2x^2 - 2x + 1. To find its derivative, we differentiate each term with respect to x.
f'(x) = -2(2x) - 2 = -4x - 2.
Step 2: Evaluate the derivative at the given x-value.
To find the slope at a specific point, we need to substitute the x-coordinate of the point into the derivative. The given point is (-1, -2), so we substitute x = -1 into f'(x).
f'(-1) = -4(-1) - 2 = 4 - 2 = 2.
Step 3: Calculate the slope of the tangent line using the derivative.
The slope of the tangent line is given by the value of the derivative at the given point, which we found to be 2.
Slope = 2.
Step 4: Use the point-slope form of a line to find the equation of the tangent line.
Now that we know the slope (m = 2) and have a point on the line (-1, -2), we can use the point-slope form of a line to find the equation of the tangent line.
The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope.
Substituting the values from the given point and the slope into the point-slope form, we get:
y - (-2) = 2(x - (-1)).
Simplifying, we get:
y + 2 = 2(x + 1).
Expanding the right side, we have:
y + 2 = 2x + 2.
Moving the constant term to the other side, we obtain:
y = 2x + 2 - 2.
Simplifying further, we get:
y = 2x.
Therefore, the equation of the tangent line to the graph of the function f(x) = -2x^2 - 2x + 1 at the point (-1, -2) is y = 2x.