2.6 (a) independent practice & hw
1. the height of a firework at an amusement park celebration can be modeled by a quadratic function. suppose the firework is launched from a platform 2 feet off the ground at a velocity of 96 feet per second.
a. write a function to represent this situation.
b. find and interpret the y-intercept in the context of this situation. c. find and interpret the axis of symmetry in the context of this situation.
d. find and interpret the vertex in the context of this situation. e. find and interpret the x-intercept(s) in the context of this situation.
2. malik is participating in a diving championship. for each of his dives, his height above the water can be modeled by a quadratic function. the diving board is 7.5 meters above the water; malik jumps with a velocity of 4.18 meters per second.
a. write a function to represent this situation.
b. find and interpret the y-intercept in the context of this situation. c. find and interpret the axis of symmetry in the context of this situation.
d. find and interpret the vertex in the context of this situation. e. find and interpret the x-intercept(s) in the context of this situation.

Question

2.6 (a) independent practice & hw
1. the height of a firework at an amusement park celebration can be modeled by a quadratic function. suppose the firework is launched from a platform 2 feet off the ground at a velocity of 96 feet per second.
 a. write a function to represent this situation.
 b. find and interpret the y-intercept in the context of this situation.  c. find and interpret the axis of symmetry in the context of this situation.
 d. find and interpret the vertex in the context of this situation.  e. find and interpret the x-intercept(s) in the context of this situation.
2. malik is participating in a diving championship. for each of his dives, his height above the water can be modeled by a quadratic function. the diving board is 7.5 meters above the water; malik jumps with a velocity of 4.18 meters per second.
 a. write a function to represent this situation.
 b. find and interpret the y-intercept in the context of this situation.  c. find and interpret the axis of symmetry in the context of this situation.
 d. find and interpret the vertex in the context of this situation.  e. find and interpret the x-intercept(s) in the context of this situation.

Accepted Answer

### Problem 1a: Write the function for the firework's height
# Explanation:
## Step1: Recall the projectile motion formula
The height $ h(t) $ of an object in projectile motion is given by the quadratic function $ h(t) = -16t^2 + v_0t + h_0 $, where $ v_0 $ is the initial velocity (in feet per second) and $ h_0 $ is the initial height (in feet).
## Step2: Identify $ v_0 $ and $ h_0 $
Here, $ v_0 = 96 $ ft/s (initial velocity) and $ h_0 = 2 $ ft (initial height from the platform).
## Step3: Substitute into the formula
Substitute $ v_0 = 96 $ and $ h_0 = 2 $ into $ h(t) = -16t^2 + v_0t + h_0 $. So the function is $ h(t) = -16t^2 + 96t + 2 $.
# Answer:
$ h(t) = -16t^2 + 96t + 2 $

### Problem 1b: Find and interpret the y - intercept
# Explanation:
## Step1: Recall the y - intercept definition
The y - intercept of a function $ y = f(t) $ is the value of $ y $ when $ t = 0 $.
## Step2: Substitute $ t = 0 $ into the function
For $ h(t)=-16t^2 + 96t + 2 $, when $ t = 0 $, $ h(0)=-16(0)^2+96(0)+2 = 2 $.
## Step3: Interpret the y - intercept
In the context of the firework, $ t = 0 $ represents the time when the firework is launched. So the y - intercept $ (0, 2) $ means that at the moment the firework is launched ($ t = 0 $ seconds), its height is 2 feet (which is the height of the platform from which it is launched).
# Answer:
The y - intercept is $ 2 $. Interpretation: At $ t = 0 $ (launch time), the firework's height is 2 feet (the height of the launch platform).

### Problem 1c: Find and interpret the axis of symmetry
# Explanation:
## Step1: Recall the formula for the axis of symmetry of a quadratic function
For a quadratic function in the form $ y = ax^2+bx + c $, the axis of symmetry is given by $ t=-\frac{b}{2a} $.
## Step2: Identify $ a $ and $ b $ from $ h(t)=-16t^2 + 96t + 2 $
Here, $ a=-16 $ and $ b = 96 $.
## Step3: Calculate the axis of symmetry
Substitute $ a=-16 $ and $ b = 96 $ into $ t =-\frac{b}{2a} $. So $ t=-\frac{96}{2\times(-16)}=-\frac{96}{-32}=3 $.
## Step4: Interpret the axis of symmetry
The axis of symmetry $ t = 3 $ means that the firework reaches its maximum height at $ t = 3 $ seconds. The parabola (graph of the height function) is symmetric about the line $ t = 3 $. So the time before $ t = 3 $ seconds and the time after $ t = 3 $ seconds (at equal distances from $ t = 3 $) will have the same height.
# Answer:
The axis of symmetry is $ t = 3 $. Interpretation: The firework reaches its maximum height at $ t = 3 $ seconds, and the height function is symmetric about $ t = 3 $.

### Problem 1d: Find and interpret the vertex
# Explanation:
## Step1: Recall the vertex formula
For a quadratic function $ y = ax^2+bx + c $, the vertex has an x - coordinate (in our case, t - coordinate) of $ t=-\frac{b}{2a} $ (which we already found as $ t = 3 $ in part c), and the y - coordinate (h - coordinate) is found by substituting $ t = 3 $ into the function $ h(t) $.
## Step2: Substitute $ t = 3 $ into $ h(t) $
$ h(3)=-16(3)^2+96(3)+2=-16\times9 + 288+2=-144 + 288+2=146 $.
## Step3: Interpret the vertex
The vertex is $ (3, 146) $. In the context of the firework, this means that at $ t = 3 $ seconds, the firework reaches its maximum height of 146 feet.
# Answer:
The vertex is $ (3, 146) $. Interpretation: At $ t = 3 $ seconds, the firework reaches its maximum height of 146 feet.

### Problem 1e: Find and interpret the x - intercept(s)
# Explanation:
## Step1: Recall the x - intercept definition
The x - intercepts of the function $ h(t) $ are the values of $ t $ when $ h(t)=0 $, i.e., we need to solve the equation $ -16t^2+96t + 2=0 $.
## Step2: Solve the quadratic equation
First, multiply through by - 1 to get $ 16t^2-96t - 2 = 0 $. Use the quadratic formula $ t=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} $, where $ a = 16 $, $ b=-96 $, and $ c=-2 $.

The discriminant $ D=b^2-4ac=(-96)^2-4\times16\times(-2)=9216 + 128 = 9344 $.

$ \sqrt{D}=\sqrt{9344}\approx96.66 $

$ t=\frac{96\pm96.66}{32} $

We have two solutions:

$ t_1=\frac{96 + 96.66}{32}=\frac{192.66}{32}\approx6.02 $

$ t_2=\frac{96 - 96.66}{32}=\frac{-0.66}{32}\approx - 0.02 $

Since time cannot be negative in this context, we consider $ t\approx6.02 $ seconds.
## Step3: Interpret the x - intercept
The x - intercept (we consider the positive one) is at $ t\approx6.02 $ seconds. In the context of the firework, this means that the firework hits the ground (height $ h(t)=0 $) at approximately $ t = 6.02 $ seconds.
# Answer:
The x - intercept (positive) is at $ t\approx6.02 $ seconds. Interpretation: The firework hits the ground at approximately $ t = 6.02 $ seconds.

### Problem 2a: Write the function for Malik's dive height
# Explanation:
## Step1: Recall the projectile motion formula for metric units
In metric units, the height $ h(t) $ of an object in projectile motion is given by $ h(t)=-4.9t^2 + v_0t+h_0 $, where $ v_0 $ is the initial velocity (in meters per second) and $ h_0 $ is the initial height (in meters). The acceleration due to gravity is approximately $ 4.9\ m/s^2 $ (so the coefficient of $ t^2 $ is - 4.9).
## Step2: Identify $ v_0 $ and $ h_0 $
Here, $ v_0 = 4.18\ m/s $ (initial velocity) and $ h_0 = 7.5\ m $ (initial height of the diving board above water).
## Step3: Substitute into the formula
Substitute $ v_0 = 4.18 $ and $ h_0 = 7.5 $ into $ h(t)=-4.9t^2 + v_0t + h_0 $. So the function is $ h(t)=-4.9t^2+4.18t + 7.5 $.
# Answer:
$ h(t)=-4.9t^2 + 4.18t + 7.5 $

### Problem 2b: Find and interpret the y - intercept
# Explanation:
## Step1: Recall the y - intercept definition
The y - intercept of a function $ y = h(t) $ is the value of $ y $ when $ t = 0 $.
## Step2: Substitute $ t = 0 $ into the function
For $ h(t)=-4.9t^2+4.18t + 7.5 $, when $ t = 0 $, $ h(0)=-4.9(0)^2+4.18(0)+7.5 = 7.5 $.
## Step3: Interpret the y - intercept
In the context of Malik's dive, $ t = 0 $ represents the time when Malik starts his dive. So the y - intercept $ (0, 7.5) $ means that at the moment Malik starts his dive ($ t = 0 $ seconds), his height above water is 7.5 meters (which is the height of the diving board).
# Answer:
The y - intercept is $ 7.5 $. Interpretation: At $ t = 0 $ (start of the dive), Malik's height above water is 7.5 meters (the height of the diving board).

### Problem 2c: Find and interpret the axis of symmetry
# Explanation:
## Step1: Recall the formula for the axis of symmetry
For a quadratic function $ y = ax^2+bx + c $, the axis of symmetry is $ t =-\frac{b}{2a} $.
## Step2: Identify $ a $ and $ b $ from $ h(t)=-4.9t^2+4.18t + 7.5 $
Here, $ a=-4.9 $ and $ b = 4.18 $.
## Step3: Calculate the axis of symmetry
$ t=-\frac{4.18}{2\times(-4.9)}=\frac{4.18}{9.8}\approx0.43 $ seconds.
## Step4: Interpret the axis of symmetry
The axis of symmetry $ t\approx0.43 $ seconds means that Malik reaches his maximum height at approximately $ t = 0.43 $ seconds. The parabola (graph of the height function) is symmetric about the line $ t\approx0.43 $. So the time before $ t\approx0.43 $ seconds and the time after $ t\approx0.43 $ seconds (at equal distances from $ t\approx0.43 $) will have the same height.
# Answer:
The axis of symmetry is $ t\approx0.43 $ seconds. Interpretation: Malik reaches his maximum height at approximately $ t = 0.43 $ seconds, and the height function is symmetric about $ t\approx0.43 $.

### Problem 2d: Find and interpret the vertex
# Explanation:
## Step1: Recall the vertex formula
The t - coordinate of the vertex is $ t =-\frac{b}{2a}\approx0.43 $ (from part c). To find the h - coordinate, substitute $ t\approx0.43 $ into $ h(t) $.
## Step2: Substitute $ t\approx0.43 $ into $ h(t) $
$ h(0.43)=-4.9(0.43)^2+4.18(0.43)+7.5 $

First, calculate $ (0.43)^2 = 0.1849 $

$ -4.9\times0.1849\approx - 0.906 $

$ 4.18\times0.43\approx1.8 $

$ h(0.43)\approx - 0.906+1.8 + 7.5=8.394\approx8.39 $ meters.
## Step3: Interpret the vertex
The vertex is approximately $ (0.43, 8.39) $. In the context of Malik's dive, this means that at approximately $ t = 0.43 $ seconds, Malik reaches his maximum height of approximately 8.39 meters.
# Answer:
The vertex is approximately $ (0.43, 8.39) $. Interpretation: At approximately $ t = 0.43 $ seconds, Malik reaches his maximum height of approximately 8.39 meters.

### Problem 2e: Find and interpret the x - intercept(s)
# Explanation:
## Step1: Recall the x - intercept definition
The x - intercepts of the function $ h(t) $ are the values of $ t $ when $ h(t)=0 $, i.e., we need to solve the equation $ -4.9t^2+4.18t + 7.5 = 0 $.
## Step2: Solve the quadratic equation
Use the quadratic formula $ t=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} $, where $ a=-4.9 $, $ b = 4.18 $, and $ c = 7.5 $.

The discriminant $ D=b^2-4ac=(4.18)^2-4\times(-4.9)\times7.5=17.4724 + 147=164.4724 $

$ \sqrt{D}=\sqrt{164.4724}\approx12.825 $

$ t=\frac{-4.18\pm12.825}{2\times(-4.9)} $

We have two solutions:

$ t_1=\frac{-4.18 + 12.825}{-9.8}=\frac{8.645}{-9.8}\approx - 0.88 $

$ t_2=\frac{-4.18-12.825}{-9.8}=\frac{-17.005}{-9.8}\approx1.735 $

Since time cannot be negative in this context, we consider $ t\approx1.74 $ seconds.
## Step3: Interpret the x - intercept
The x - intercept is at $ t\approx1.74 $ seconds. In the context of Malik's dive, this means that Malik hits the water (height $ h(t)=0 $) at approximately $ t = 1.74 $ seconds.
# Answer:
The x - intercept (positive) is at $ t\approx1.74 $ seconds. Interpretation: Malik hits the water at approximately $ t = 1.74 $ seconds.

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QUESTION IMAGE

Question

Explanation:

Problem 1a: Write the function for the firework's height

Step1: Recall the projectile motion formula

Step2: Identify \( v_0 \) and \( h_0 \)

Step3: Substitute into the formula

Step1: Recall the y - intercept definition

Step2: Substitute \( t = 0 \) into the function

Step3: Interpret the y - intercept

Step1: Recall the formula for the axis of symmetry of a quadratic function

Step2: Identify \( a \) and \( b \) from \( h(t)=-16t^2 + 96t + 2 \)

Step3: Calculate the axis of symmetry

Step4: Interpret the axis of symmetry

Answer:

Problem 1b: Find and interpret the y - intercept