Question

1. the function $f(x) = \frac{1}{2}x - 6$ was replaced with $f(x + k) = \frac{1}{2}x - 4$. what is the value of $k$?
2. avis used a quadratic function to solve a problem. the factored form of the function is show below.

\\((4x + 3)(6x - 3) = 0\\)
what is the positive solution to the problem?

1. the average daily high temperature for the month of may in ocala, florida is approximated by the function $f(n) = 0.2n + 80$, where $n$ is the day of the month. may has 31 days. the maximum daily high temperature occurred on may $31^{st}$. what was the maximum temperature?

Explanation:

(Problem 10):

Step1: Define $f(x+k)$

Substitute $x+k$ into $f(x)$:
$f(x+k)=\frac{1}{2}(x+k)-6$

Step2: Set equal to given expression

Equate to $\frac{1}{2}x - 4$:
$\frac{1}{2}(x+k)-6=\frac{1}{2}x - 4$

Step3: Simplify and solve for $k$

Expand left side: $\frac{1}{2}x+\frac{1}{2}k-6=\frac{1}{2}x - 4$
Cancel $\frac{1}{2}x$, solve: $\frac{1}{2}k=2$
$k=4$

(Problem 12):

Step1: Apply zero product property

Set each factor to 0:
$4x+3=0$ or $6x-3=0$

Step2: Solve each equation

For $4x+3=0$: $x=-\frac{3}{4}$
For $6x-3=0$: $6x=3$ → $x=\frac{1}{2}$

Step3: Select positive solution

Choose the positive value: $x=\frac{1}{2}$

(Problem 14):

Step1: Substitute $n=31$ into $f(n)$

$f(31)=0.2(31)+80$

Step2: Calculate the value

Compute $0.2\times31=6.2$, then add 80:
$6.2+80=86.2$

Answer:

1. For problem 10: $k=4$
2. For problem 12: $x=\frac{1}{2}$
3. For problem 14: $86.2$