Question

The fuel efficiency of a vehicle can be modeled with a quadratic function over a limited span

of typical highway speeds where x is the speed in mph and y is the fuel efficiency in mpg.
Suppose that a certain car has its best efficiency, 34 miles per gallon, at a speed of 42 miles
per hour. At 70 miles per hour, the efficiency is reduced to 20 miles per gallon.
a) Find a quadratic function that models the car's efficiency.
Hint: Use the vertex form of y = a(x - h)² + k, where (h, k) is the vertex and (x, y) is
another point on the graph.

Answer 1

The equation is modeled in vertex form to be y = -1/56 (x - 42)² + 34

How to model the quadratic equation

To find a quadratic function that models the car's efficiency, we can use the vertex form of a quadratic function, which is given by:

y = a(x - h)²

where (h, k) is the vertex of the parabola.

Given that the car best efficiency is 34 miles per gallon at a speed of 42 miles per hour. This is the vertex (h, k) = (42, 34)

The efficiency is reduced to 20 miles per gallon at 70 miles per hour, we have two points on the graph: (42, 34) and (70, 20).

substitute for the vertex

y = a(x - 42)² + 34

use the second point (70, 20) to find a:

20 = a (70 - 42)² + 34

20 = a (28)² + 34

20 = 784a + 34

784a = -14

a = -14/784

a = -1/56

y = -1/56 (x - 42)² + 34