Question

The function y=x²−4x+8 approximates the height, y, of a bird, and its horizontal distance, x, as it flies from one fence post to another. All distances are in feet. Complete the square to find and interpret the extreme value (vertex).

a) Find the completed square form of the quadratic equation.

b) Identify the coordinates of the vertex (extreme value).

c) Interpret the meaning of the vertex in the con of the bird's flight.

d) Determine the maximum or minimum height represented by the vertex.

Answer 1

To complete the square for the quadratic equation y = x² - 4x + 8, we take half the coefficient of x, square it, and incorporate it into the equation. The vertex (extreme value) of the quadratic function is (2, 4), which represents the maximum height of the bird. The maximum height represented by the vertex is 4 feet.

Step-by-step explanation:

To complete the square for the quadratic equation y = x² - 4x + 8, we follow these steps:

1. Take half the coefficient of x (-4/2 = -2) and square it to get 4.
2. Add the result from step 1 as a constant term inside the parentheses, and subtract it outside the parentheses. y = (x² - 4x + 4) + 8 - 4.
3. Simplify the equation: y = (x - 2)² + 4.

(b) The coordinates of the vertex (extreme value) can be found by identifying the values of x and y in the completed square form. In this case, the vertex is located at (2, 4).

(c) The vertex represents the highest point in the bird's flight trajectory. It indicates the maximum height the bird reaches as it flies from one fence post to another.

(d) The maximum height represented by the vertex is 4 feet.