Intermediate Algebra

# Quadratic Functions – Stopping Distance and Velocity

When working with quadratic functions, as with other function types, two primary questions we will ask about the function variables and how they interact are:

• Given a particular input value, what is the corresponding output value?
• Given a particular output value, what is the corresponding input value?

As a student of this material, if you can recognize if one of these questions is being asked and which one is being asked, then are are a step closer to finding the solution.

EXAMPLE: Suppose you are iven the quadratic function d(x) = 0.044x^2 + 1.1x where x represents the velocity of a vehicle in mph and d represents the stopping distance of that vehicle in feet.

a) If the vehicle is traveling 45 miles per hour, how many feet are required for it to stop?

SOLUTION: This is a “given input find output” question. Since we know the input, we evaluate the function at the input value to directly compute the output as follows:

d(30) = 0.044(45)^2 + 1.1(45) = 138.6

Meaning: When a vehicle is traveling 45 mph, 138.6 feet  are needed for it to stop.

b) If the vehicle takes 75 feet to stop, how fast is it traveling?

SOLUTION: This is a “given output find input” question. We are given the value for the output which means we must set up and solve the following equation (note that the 75 value replaces d(x)):

75 = 0.044x^2 + 1.1x

There are many ways to solve this equation (note: I do NOT advocate guess and check as a primary method due to issues of reliability and efficiency). I am going to use the method here of graphing and finding the intersection (TI83/84 calculator).

Step 1: Pres Y= and enter y1 = 75 and y2 = 0.044x^2 + 1.1x on your calculator

Step 2: Press Window and enter xmin=0, xmax=50, ymin=0, ymax=100.

Step 3: Press GRAPH and be sure that the intersection of the two graphs above appears in your window. Your graph should look similar to the following (without the labels on your axes):

Find the intersection. The x-value of the intersection is the solution to the equation you are trying to solve.

Step 4: 2nd>Trace>5: Intersect then press Enter three times until you see Intersection in the window bottom.

Result: Intersection point is identified as (30.64, 75) rounded to two decimals. Therefore the solution to the initial equation is x = 30.64.

Meaning: When a vehicle needs 75 feet to stop, it is traveling at 30.64 mph.