3. complete the square to write the function in vertex - form, (y = a(x - h)^2 + k), and identify the vertex.
a) (y - 1=x^{2}-10x)
b) (y = 4x^{2}+24x + 13)
4. same as one of the problems in wiley find the minimum value of the function (h(x)=x^{2}-10x + 7), if it has one.
5. similar to one of the problems in wiley explain why (k(x)=-(x + 8)^{2}+19) has no minimum.
6. a carpenter finds that if she charges (p) dollars for a chair, she sells (1200 - 3p) of them each year.
a) what happens when she charges $400 for a chair? should she charge $400 or more for a chair? explain why or why not.
b) construct the revenue function, (r(p)). is it quadratic?
c) sketch a graph of the revenue function, (r(p)). label your axes. hint: use the factored form to find the zeros and then find the vertex.
d) how much should she charge to maximize her annual revenue?
e) what is her maximum annual revenue?

Question

3. complete the square to write the function in vertex - form, (y = a(x - h)^2 + k), and identify the vertex.
 a) (y - 1=x^{2}-10x)
 b) (y = 4x^{2}+24x + 13)
4. same as one of the problems in wiley find the minimum value of the function (h(x)=x^{2}-10x + 7), if it has one.
5. similar to one of the problems in wiley explain why (k(x)=-(x + 8)^{2}+19) has no minimum.
6. a carpenter finds that if she charges (p) dollars for a chair, she sells (1200 - 3p) of them each year.
 a) what happens when she charges $400 for a chair? should she charge $400 or more for a chair? explain why or why not.
 b) construct the revenue function, (r(p)). is it quadratic?
 c) sketch a graph of the revenue function, (r(p)). label your axes. hint: use the factored form to find the zeros and then find the vertex.
 d) how much should she charge to maximize her annual revenue?
 e) what is her maximum annual revenue?

Accepted Answer

### 3a.
# Answer:
$y=(x - 5)^2+24$, vertex: $(5,24)$
# Explanation:
## Step1: Isolate y
$y=x^{2}-10x + 1$
## Step2: Complete the square
For the quadratic $x^{2}-10x$, the coefficient of $x$ is $- 10$. Half of it is $-5$, and its square is $25$. So $y=(x^{2}-10x + 25)-25 + 1$.
## Step3: Write in vertex - form
$y=(x - 5)^2-24$. Since $y - 1=x^{2}-10x$, then $y=(x - 5)^2+24$. The vertex of the parabola $y=a(x - h)^2+k$ is $(h,k)$, so the vertex is $(5,24)$.

### 3b.
# Answer:
$y = 4(x+3)^2 - 11$, vertex: $(-3,-11)$
# Explanation:
## Step1: Factor out the coefficient of $x^{2}$
$y=4(x^{2}+6x)+13$
## Step2: Complete the square inside the parentheses
For $x^{2}+6x$, half of the coefficient of $x$ is $3$ and its square is $9$. So $y=4(x^{2}+6x + 9)-4	imes9+13$.
## Step3: Write in vertex - form
$y = 4(x + 3)^2-36 + 13=4(x + 3)^2-11$. The vertex is $(-3,-11)$.

### 4.
# Answer:
$-18$
# Explanation:
## Step1: Complete the square for $h(x)$
$h(x)=x^{2}-10x + 7=(x^{2}-10x+25)-25 + 7$.
## Step2: Write in vertex - form
$h(x)=(x - 5)^2-18$. Since the coefficient of $(x - 5)^2$ is $1>0$, the parabola opens upward and the minimum value occurs at the vertex. The minimum value of $h(x)$ is $-18$.

### 5.
# Answer:
The function $k(x)=-(x + 8)^2+19$ has no minimum.
# Explanation:
## Step1: Analyze the form of the quadratic function
The function $k(x)=-(x + 8)^2+19$ is in the vertex - form $y=a(x - h)^2+k$, where $a=-1<0$.
## Step2: Determine the shape of the parabola
Since $a=-1<0$, the parabola opens downward. So it has a maximum value (at the vertex $( - 8,19)$) and no minimum value.

### 6a.
# Answer:
When $p = 400$, the number of chairs sold is $1200-3	imes400=0$. She should not charge $400$ or more for a chair.
# Explanation:
## Step1: Calculate the number of chairs sold
The number of chairs sold $n = 1200-3p$. When $p = 400$, $n=1200-3	imes400=1200 - 1200=0$. If she charges $400$ or more, $n\leq0$, so she will not sell any chairs.

### 6b.
# Answer:
$R(p)=p(1200 - 3p)=-3p^{2}+1200p$, it is quadratic.
# Explanation:
## Step1: Construct the revenue function
Revenue $R(p)$ is the price per unit $p$ times the number of units sold. The number of units sold is $1200 - 3p$. So $R(p)=p(1200 - 3p)$.
## Step2: Expand the function
$R(p)=1200p-3p^{2}=-3p^{2}+1200p$. Since it is in the form $y = ax^{2}+bx + c$ ($a=-3$, $b = 1200$, $c = 0$), it is a quadratic function.

### 6c.
# Answer:
Zeros: $p = 0$ and $p = 400$, vertex: $(200,120000)$.
# Explanation:
## Step1: Find the zeros
Set $R(p)=-3p^{2}+1200p=0$. Factor out $-3p$: $-3p(p - 400)=0$. So the zeros are $p = 0$ and $p = 400$.
## Step2: Find the vertex
The $x$ - coordinate of the vertex of a quadratic function $y = ax^{2}+bx + c$ is $x=-\frac{b}{2a}$. For $R(p)=-3p^{2}+1200p$, $a=-3$ and $b = 1200$. So $p=-\frac{1200}{2	imes(-3)} = 200$. Substitute $p = 200$ into $R(p)$: $R(200)=-3	imes200^{2}+1200	imes200=-3	imes40000+240000=120000$.

### 6d.
# Answer:
$200$ dollars
# Explanation:
## Step1: Recall the vertex of the revenue function
The revenue function $R(p)=-3p^{2}+1200p$ is a quadratic function. The $p$ - value of the vertex of the parabola $y = ax^{2}+bx + c$ is $p=-\frac{b}{2a}$. Here $a=-3$ and $b = 1200$, so $p = 200$.

### 6e.
# Answer:
$120000$ dollars
# Explanation:
## Step1: Substitute the optimal price into the revenue function
We found that the optimal price $p = 200$. Substitute $p = 200$ into $R(p)=-3p^{2}+1200p$. $R(200)=-3	imes200^{2}+1200	imes200=-120000 + 240000=120000$.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

3a.

Step1: Isolate y

Step2: Complete the square

Step3: Write in vertex - form

3b.

Step1: Factor out the coefficient of $x^{2}$

Step2: Complete the square inside the parentheses

Step3: Write in vertex - form

4.

Step1: Complete the square for $h(x)$

Step2: Write in vertex - form

5.

Answer: