# 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.