This week we learned applications of quadratic functions with word problems. I choose this subject because I was interested in how these quadratic lines could be applied to real life situations. Quadratic applications are important because they allow you to   learn about different ways to see math problems and numbers in general with variables.

1. Understand the Problem: Read the word problem carefully to identify what information is given and what needs to be found. Look for keywords that indicate a quadratic relationship, such as “area,” “height,” “distance,” “time,” etc.

2. Define Variables: Assign variables to the quantities mentioned in the problem. For example, if the problem talks about the area of a rectangular garden, you can let  x  represent the length and  y  represent the width unless given.

3. Turn it into a Quadratic Equation: Use the given information to formulate a quadratic equation that represents the problem. This equation will typically have the form  y = ax^2 + bx + c , where a , b , and  c  are constants.

4. Solve the Equation: Depending on the problem, you may need to solve the quadratic equation to find the values of the variables. You can use factoring, the quadratic formula, or completing the square to solve for  x  or  y .

5. Interpret the Solution: Once you have the solution, interpret it in the context of the problem. Sometimes, you may need to discard extraneous solutions that don’t make sense in the given situation.

Here’s an example to illustrate these steps:

Example Problem:A ball is thrown upward from the top of a 30-meter building with an initial velocity of 20 m/s. The height of the ball above the ground at time \( t \) seconds is given by the equation . When does the ball hit the ground?

Steps to Solve:

1. Understand the Problem: We need to find the time at which the ball hits the ground, which corresponds to when the height \( h(t) \) is 0.

2. Define Variables: Let t represent time in seconds.

3. Formulate the Quadratic Equation: The equation for the height of the ball is given as . Set \ h(t)  to 0 and solve for  t :

4.Solve the Equation: Use the quadratic formula with , , and to find the values of :

5. Interpret the Solution: Since time cannot be negative in this context, we discard the negative solution. The ball hits the ground after approximately seconds.

This process demonstrates how quadratic functions can be applied to real-world scenarios to solve problems related to motion, distance, height, and other physical quantities.