Given f ( x ) = 1 x + 4 , find the average rate of change of f ( x ) on the interval [ 5 , 5 + h ] . Your answer willbe an expression involving h .

Answer 1

So when you're asked to find "Rate of Change" for a straight line, you know you need to think about the slope of the line. Ask, "how fast do the y-coordinates change, in comparison to how fast the x-coordinates change?"

To find how fast each change is happening, both vertically (y-coordinates) and horizontally (x-coordinates), you can identify two points on the line, usually called (x₁,y₁) and (x₂,y₂), and find the differences in each coordinate: Remember "difference" means subtract, so:

(y₂-y₁) tells you how much the y-coordinates are changing, and

(x₂-x₁) tells you how much the x-coordinates are changing.

Now that we have two numbers that tell us the amount of change in each dimension, we can find the RATE of change -- how much does the y-coordinate change **per** change in x-coordinate? "Per" means divide, so let's divide those two quantities, and use the following formula:

(y₂-y₁) ÷ (x₂-x₁)

This formula is also known as the formula for rate of change, given two points on the line.

Since you don't have two full points given to you in the form of (x₁,y₁) and (x₂,y₂), your teacher is asking you for a little more work. You've been given two x-coordinates -- your x₁ and x₂ -- and you need to find the y-coordinates to go with them.

For your first x-coordinate, x₁, which is 5, you can just pop that right into your function equation: f(x) = 1x + 4, so f(5) = 1(5) + 4. Evaluate that expression, and you'll have your y₁ coordinate.

For your second x-coordinate, x₂, which is 5+h, you won't end up with a single number, but with an expression. You'll also substitute this into your function equation, just be sure that you distribute and multiply properly: f(5+h) = 1(5+h) + 4. Simplify that expression to get your y₂ coordinate.

So to recap, finding rate of change (or slope of a line) can be calculated with the expression (y₂-y₁) ÷ (x₂-x₁). The problem gives you the information you need to identify the 4 values, x₁, x₂, y₁, and y_2. Then just put your values into the equation and you can find the rate of change.