How To Find An Equation Of The Tangent Line To The Graph Of F At The Indicated Point

The derivative of a function gives the instantaneous rate of change for a given point. Think of the way the velocity of a car is always changing as it accelerates and decelerates. Although you can calculate the average velocity for the entire trip, sometimes you need to know the velocity for a particular instant. The derivative provides this information, not just for velocity but for any rate of change. A tangent line shows what could have been if the rate had been constant, or what could be if it remains unchanged.

Step 1

Determine the coordinates of the indicated point by plugging the value of x into the function. For example, to find the tangent line where x = 2 of the function F(x) = -x^2 + 3x, plug x into the function to find F(2) = 2. Thus the coordinate would be (2, 2).

Step 2

Find the derivative of the function. Think of the derivative of a function as a formula that gives the slope of the function for any value of x. For example, the derivative F'(x) = -2x + 3.

Step 3

Calculate the slope of the tangent line by plugging the value of x into the function of the derivative. For example, slope = F'(2) = -2 * 2 + 3 = -1.

Step 4

Find the y-intercept of the tangent line by subtracting the slope times the x-coordinate from the y-coordinate: y-intercept = y1 – slope x1. The coordinate found in Step 1 must satisfy the tangent line equation. Therefore plugging in the coordinate values into the slope-intercept equation for a line, you can solve for the y-intercept. For example, y-intercept = 2 – (-1 2) = 4.

Step 5

Write the equation of the tangent line in the form y = slope * x + y-intercept. In the example given, y = -x + 4.