PERIOD $\newline$ DATE $\newline$ NAME $\newline$ $3$ . The height in feet of a thrown football is modeled by the equation $f(t)=6+30 t-16 t^{2}$ , where time $t$ is measured in seconds. $\newline$ a. What does the constant $6$ mean in this situation? $\newline$ b. What does the $30 t$ mean in this situation? $\newline$ c. How do you think the squared term $-16 t^{2}$ affects the value of the function $f$ ? What does this term reveal about the situation? $\newline$ $4$ . The height in feet of an arrow is modeled by the equation $h(t)=(1+2 t)(18-8 t)$ , where $t$ is seconds after the arrow is shot. $\newline$ a. When does the arrow hit the ground? Explain or show your reasoning. $\newline$ b. From what height is the arrow shot? Explain or show your reasoning.

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Algebra 1

Interpret parts of quadratic expressions: word problems

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Q. PERIOD

\newline

DATE

\newline

NAME

\newline

3

. The height in feet of a thrown football is modeled by the equation

f(t)=6+30 t-16 t^{2}

, where time

t

is measured in seconds.

\newline

a. What does the constant

6

mean in this situation?

\newline

b. What does the

30 t

mean in this situation?

\newline

c. How do you think the squared term

-16 t^{2}

affects the value of the function

f

? What does this term reveal about the situation?

\newline

4

. The height in feet of an arrow is modeled by the equation

h(t)=(1+2 t)(18-8 t)

, where

t

is seconds after the arrow is shot.

\newline

a. When does the arrow hit the ground? Explain or show your reasoning.

\newline

b. From what height is the arrow shot? Explain or show your reasoning.

Analyzing Initial Football Height: a. Analyzing the football height equation for the constant $6$ . The equation is $f(t) = 6 + 30t - 16t^2$ . The constant $6$ represents the initial height of the football when $t = 0$ .
Interpreting Initial Upward Velocity: b. Interpreting the term $30t$ in the football height equation. $\newline$ The term $30t$ represents the initial upward velocity of the football. The coefficient $30$ indicates the rate at which the height changes with time at the initial moment of the throw.
Understanding Effect of Gravity: c. Understanding the effect of the squared term $-16t^2$ on the football height equation. $\newline$ The term $-16t^2$ represents the effect of gravity on the football's height over time. It causes the height to decrease quadratically over time, indicating that the football will eventually reach a peak and then start to fall back down.
Determining Arrow Hits Ground: a. Determining when the arrow hits the ground using the arrow height equation. $\newline$ The equation is $h(t) = (1 + 2t)(18 - 8t)$ . $\newline$ To find when the arrow hits the ground, we need to solve for $t$ when $h(t) = 0$ . $\newline$ $0 = (1 + 2t)(18 - 8t)$ $\newline$ This is a quadratic equation that can be expanded and then factored or solved using the quadratic formula. $\newline$ Expanding: $0 = 18 + 36t - 8t - 16t^2$ $\newline$ Simplifying: $0 = 18 + 28t - 16t^2$ $\newline$ Rearranging: $16t^2 - 28t - 18 = 0$ $\newline$ We can solve this quadratic equation by factoring or using the quadratic formula.
Calculating Initial Arrow Height: b. Calculating the initial height from which the arrow is shot. $\newline$ To find the initial height, we need to evaluate $h(t)$ at $t = 0$ . $\newline$ $h(0) = (1 + 2(0))(18 - 8(0))$ $\newline$ $h(0) = (1)(18)$ $\newline$ $h(0) = 18$ feet $\newline$ The arrow is shot from a height of $18$ feet.