My assignment was for each polynomial topic is to give an "example and written description." What would be an "example" and "written description" of a CONNECTION TO A QUADRATIC FUNCTIONS.
To provide an example and written description of a connection to a quadratic function, let's start by understanding what a quadratic function is. A quadratic function is a polynomial function of degree two, typically written in the form of:
f(x) = ax² + bx + c
Here, 'a', 'b', and 'c' are constants, and 'x' is the variable representing the input values.
To establish a connection to a quadratic function, let's look at the concept of projectile motion. When an object is thrown or launched into the air, its trajectory can be described using a quadratic function. The height of the object, 'h', at a given time, 't', can be modeled using a quadratic function.
An example of a connection to a quadratic function would be calculating the maximum height achieved by a projectile. Let's say we have a projectile launched from the ground with an initial velocity of 20 meters per second at an angle of 45 degrees with the horizontal.
To find the maximum height, we can use the following steps:
1. Break down the initial velocity into its horizontal and vertical components.
- The horizontal component is given by: Vx = V₀ * cos(θ), where V₀ is the initial velocity, and θ is the launch angle.
- The vertical component is given by: Vy = V₀ * sin(θ).
2. Use the equation for vertical motion to find the time taken to reach the maximum height:
- h(t) = Vyt - (1/2)gt², where h(t) represents the height at time 't', Vyt is the vertical component times 't', and 'g' is the acceleration due to gravity (approximately 9.8 m/s²).
3. Determine the time when the projectile reaches its maximum height by finding the vertex of the quadratic function.
- The vertex of a quadratic function in standard form, f(x) = ax² + bx + c, has the form (h, k), where h = -b / 2a and k = f(h).
- In our case, h represents time, and k represents the maximum height.
Once you have determined the time at which the projectile reaches its maximum height, you can substitute it back into the height equation (h(t)) to find the corresponding maximum height.
By applying the concepts of quadratic functions to projectile motion, you can calculate the maximum height, which demonstrates a connection to quadratic functions in a real-world context.