Find the slope-predictor formula and write an equation for the line tangent to (a, f(a)) for f(x) = x2 - 6x + 4. Make sure you detail the process.
To find the slope-predictor formula and write an equation for the line tangent to the function at a given point (a, f(a)), we need to follow a step-by-step process:
Step 1: Find the derivative of the function f(x).
The derivative of f(x) can be found using the power rule and the sum rule of derivatives:
f'(x) = 2x - 6
Step 2: Evaluate the derivative at the given x-coordinate, a.
Substituting a into the derivative equation, we obtain:
f'(a) = 2a - 6
Step 3: Determine the slope of the line tangent to the function at the point (a, f(a)).
The slope of the tangent line is given by the derivative evaluated at the x-coordinate, a. Therefore, the slope of the tangent line is:
m = f'(a) = 2a - 6
Step 4: Write the equation of the tangent line.
The equation of a line is generally represented as y = mx + b, where m is the slope and b is the y-intercept. Since we want to write the equation in terms of the point (a, f(a)), we substitute a for x and f(a) for y:
f(a) = (2a - 6)a + b
Step 5: Solve for the y-intercept, b.
To find the value of b, we substitute the x-coordinate, a, and the y-coordinate, f(a), into the equation:
f(a) = (2a - 6)a + b
f(a) = 2a^2 - 6a + b
Since f(a) = a^2 - 6a + 4, we can set the two expressions equal to each other:
a^2 - 6a + 4 = 2a^2 - 6a + b
Step 6: Simplify and solve for b.
By comparing the coefficients of each term on both sides of the equation, we can conclude that b = 4.
Step 7: Write the final equation of the tangent line.
Using the determined values of m and b, we can write the equation of the tangent line:
y = (2a - 6)x + 4
Thus, the equation for the line tangent to the function f(x) = x^2 - 6x + 4 at the point (a, f(a)) is y = (2a - 6)x + 4.