Lesson 17
Quadratic Meanings
- Let’s explore the meaning of quadratics.
17.1: Area Between Triangles
The area of the shaded region from the image can be represented by the expression \frac{1}{2}(2+2a)(2+2a) - \frac{1}{2}\boldcdot2^2 which can be rearranged to 2a^2 + 4a. To find the value of a when the shaded area is 30 square centimeters, Mai sets up the equation 2a^2 + 4a = 30.
- One solution to the equation is a = \text{-}5. Find another solution. Explain or show your reasoning.
- What do the 2 solutions to the equation represent in this situation? Do the values make sense?
17.2: Getting the Ball Off the Roof
A ball is kicked off the roof of a building so that its height above the ground, given in feet, t seconds after it is kicked is represented by the equation h(t) = \text{-}16t^2 + 33t + 37.
- At what height is the ball when it is kicked? Explain or show your reasoning.
- At what height is the ball 2 seconds after it is kicked? Explain or show your reasoning.
- What does it mean for the situation when h(t) = 8?
- What does it mean for the situation when t = 1.3?
- Graph the function.
- Approximate the number of seconds after the ball is kicked when it will hit the ground. Explain how you know.
- Approximate the number of seconds after the ball is kicked when it will reach its highest point. Explain how you know.
- Approximate the number of seconds after the ball is kicked when it will reach its starting height again. Explain how you know.
- Write an equation that represents the exact moment when the ball hits the ground.
17.3: Kicking the Field Goal
Andre kicks a football for a field goal. The height above ground, given in feet, t seconds after it is kicked, is represented by the equation g(t)=\text{-}16t^2+56t+0.5.
- At what height is the ball when it is kicked? Explain or show your reasoning.
- At what height is the ball 2 seconds after it is kicked?
- What does it mean for the situation when g(t)=10?
- What does it mean for the situation when t=1.7?
- Graph the function.
- Approximate the number of seconds after the ball is kicked when it will hit the ground. Explain how you know.
- Approximate the number of seconds after the ball is kicked when it will reach its highest point. Explain how you know.
- Approximate the number of seconds after the ball is kicked when it will be 10 feet above the ground for the second time. Explain how you know.
- Write an equation that would give the exact time when the ball is 10 feet above the ground.