10. roller coaster the graph shows the height of a roller coaster in terms of its distance away from the starting point. describe and interpret the end - behavior in the context of the situation. mixed exercises 11. model the height of a fish t seconds after it is thrown to a dolphin from a 64 - foot - tall platform can be modeled by the equation h(t)=-16t² + 48t + 64, where h(t) is the height of the fish in feet. the graph of the function is shown. a. estimate the t - coordinate at which the height of the fish changes from increasing to decreasing. describe the meaning in terms of the context of the situation. b. describe and interpret the end behavior of h(t) in the context of the situation. identify and estimate the x - and y - values of the extrema. round to the nearest tenth if necessary. then use the graphs to describe the end behavior of each function.

Question

Accepted Answer

### 11a
# Explanation:
## Step1: Find the vertex of the parabola
The function $h(t)=- 16t^{2}+48t + 64$ is a quadratic function in the form $y = ax^{2}+bx + c$ where $a=-16$, $b = 48$, $c = 64$. The $t$-coordinate of the vertex of a quadratic function $y=ax^{2}+bx + c$ is given by $t=-\frac{b}{2a}$.
$t=-\frac{48}{2	imes(-16)}=\frac{48}{32}=\frac{3}{2}=1.5$
## Step2: Interpret the result
The $t$-coordinate at which the height of the fish changes from increasing to decreasing is $t = 1.5$ seconds. This means that 1.5 seconds after the fish is thrown, it reaches its maximum height.

# Answer:
The $t$-coordinate is $1.5$ seconds. It represents the time when the fish reaches its maximum height.

### 11b
# Explanation:
## Step1: Analyze the end - behavior
As $t	o+\infty$, $h(t)=-16t^{2}+48t + 64$. Since the leading term $-16t^{2}$ has a negative coefficient and the degree of the polynomial is 2 (even), $\lim_{t	o+\infty}h(t)=-\infty$.
In the context of the situation, as time $t$ goes on (a long time after the fish is thrown), the height of the fish will approach the ground (height $h(t) = 0$ and then go below 0 in a non - physical sense for the model, since in reality it will hit the ground and stop falling).

# Answer:
As $t	o+\infty$, $h(t)	o-\infty$. This means that over time, the fish will fall towards the ground and eventually hit it.

### 12
# Explanation:
## Step1: Find extrema
For the given graph, the local maximum occurs around $x=-2$ with $y\approx8$ and the local minimum occurs around $x = 2$ with $y\approx - 8$.
## Step2: Analyze end - behavior
As $x	o-\infty$, $y	o-\infty$ and as $x	o+\infty$, $y	o-\infty$.

# Answer:
Local maximum: $x\approx - 2,y\approx8$; Local minimum: $x\approx2,y\approx - 8$. End - behavior: As $x	o-\infty$, $y	o-\infty$ and as $x	o+\infty$, $y	o-\infty$.

### 13
# Explanation:
## Step1: Find extrema
The local minimum occurs around $x = 0$ with $y\approx - 2$ and local maxima occur around $x=-4$ with $y\approx4$ and $x = 4$ with $y\approx4$.
## Step2: Analyze end - behavior
As $x	o-\infty$, $y	o-\infty$ and as $x	o+\infty$, $y	o-\infty$.

# Answer:
Local minima: $x\approx0,y\approx - 2$; Local maxima: $x\approx - 4,y\approx4,x\approx4,y\approx4$. End - behavior: As $x	o-\infty$, $y	o-\infty$ and as $x	o+\infty$, $y	o-\infty$.

### 14
# Explanation:
## Step1: Find extrema
The local maximum occurs around $x = 0$ with $y\approx8$ and local minima occur around $x=-2$ with $y\approx - 2$ and $x = 2$ with $y\approx - 2$.
## Step2: Analyze end - behavior
As $x	o-\infty$, $y	o-\infty$ and as $x	o+\infty$, $y	o-\infty$.

# Answer:
Local maximum: $x\approx0,y\approx8$; Local minima: $x\approx - 2,y\approx - 2,x\approx2,y\approx - 2$. End - behavior: As $x	o-\infty$, $y	o-\infty$ and as $x	o+\infty$, $y	o-\infty$.

### 15
# Explanation:
## Step1: Find extrema
The local minimum occurs around $x=-2$ with $y\approx - 4$ and local maxima occur around $x = 0$ with $y\approx4$ and $x = 4$ with $y\approx4$.
## Step2: Analyze end - behavior
As $x	o-\infty$, $y	o+\infty$ and as $x	o+\infty$, $y	o+\infty$.

# Answer:
Local minimum: $x\approx - 2,y\approx - 4$; Local maxima: $x\approx0,y\approx4,x\approx4,y\approx4$. End - behavior: As $x	o-\infty$, $y	o+\infty$ and as $x	o+\infty$, $y	o+\infty$.

### 16
# Explanation:
## Step1: Find extrema
The local minimum occurs around $x = 0$ with $y\approx - 2$ and the local maximum occurs around $x=-2$ and $x = 2$ with $y\approx10$.
## Step2: Analyze end - behavior
As $x	o-\infty$, $y	o+\infty$ and as $x	o+\infty$, $y	o+\infty$.

# Answer:
Local minimum: $x\approx0,y\approx - 2$; Local maxima: $x\approx - 2,y\approx10,x\approx2,y\approx10$. End - behavior: As $x	o-\infty$, $y	o+\infty$ and as $x	o+\infty$, $y	o+\infty$.

### 17
# Explanation:
## Step1: Find extrema
The local minimum occurs around $x = 0$ with $y\approx - 2$ and local maxima occur around $x=-2$ with $y\approx70$ and $x = 2$ with $y\approx70$.
## Step2: Analyze end - behavior
As $x	o-\infty$, $y	o+\infty$ and as $x	o+\infty$, $y	o+\infty$.

# Answer:
Local minimum: $x\approx0,y\approx - 2$; Local maxima: $x\approx - 2,y\approx70,x\approx2,y\approx70$. End - behavior: As $x	o-\infty$, $y	o+\infty$ and as $x	o+\infty$, $y	o+\infty$.

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QUESTION IMAGE

Question

Explanation:

11a

Step1: Find the vertex of the parabola

Step2: Interpret the result

Step1: Analyze the end - behavior

Step1: Find extrema

Step2: Analyze end - behavior

Answer:

11b