Math / Algebra

Question```latex \textbf{Practice Q\#2: Determine if the given scenarios are linear relations AND functions.}
\begin{enumerate} \item A campsite in a Provincial park charges \ $35 for the car + driver, then \$15 per additional person. \\ Linear$ (y / n) $: \qquad Function$ (y / n) : \qquad Reason: \item A population of fruit flies doubles every 12 hours. \\ Linear (y / n) $: \qquad Function$ (y / n) : \qquad Reason: \item Linear (y / n) $: \qquad y \\ Function$ (y / n) : \qquad Reason: \item \begin{tabular}{|c|c|c|c|c|} \hline s(\mathrm{~km} / \mathrm{h}) & 50 & 60 & 70 & 80 \\ \hline d(\mathrm{~m}) & 13 & 20 & 27 & 35 \\ \hline \end{tabular} \begin{tabular}{|c|c|c|c|c|} \hline t(\min ) & 0 & 2 & 4 & 6 \\ \hline a(m)$ & 12000 & 11600 & 11200 & 10800 \\ \hline \end{tabular} \end{enumerate}
\textbf{Practice Q\#3: Determine the rate of change for each segment of Kat's ocean dive.}
\begin{tabular}{|c|c|} \hline Segment & Rate of Change \\ \hline OA & \\ \hline AB & \\ \hline BC & \\ \hline CD & \\ \hline DE & \\ \hline \end{tabular} ```

Studdy Solution

STEP 1

What is this asking? We need to figure out if some scenarios are linear relationships and if they're functions, and then find the rate of change for different parts of Kat's ocean dive. Watch out! Don't mix up linear relationships with functions–something can be one but not the other!
Also, be careful with units when calculating rates of change.

STEP 2

1. Campsite Costs
2. Fruit Fly Population
3. X-Shaped Graph
4. Parabolic Graph
5. Kat's Dive

STEP 3

Let's define our variables! $c$ is the total cost and $p$ is the number of additional people.

STEP 4

The cost is $\$35$ plus $\$15$ per extra person.
We can write this as $c = 35 + 15 \cdot p$ .

STEP 5

This is a linear relationship because it's in the form $y = mx + b$ , where $m = 15$ and $b = 35$ .
It's also a function because each input ( $p$ ) has exactly one output ( $c$ ).

STEP 6

Let $P$ be the population and $t$ be the time in hours.

STEP 7

Doubling every 12 hours means we multiply by 2 every 12 hours.
This can be written as $P(t) = P_0 \cdot 2^{\frac{t}{12}}$ , where $P_0$ is the initial population.

STEP 8

This is not linear because of the exponent.
It's an exponential relationship.
It is a function because each time has one population.

STEP 9

This graph is made of two intersecting lines, so it's not a single linear relationship.

STEP 10

It's not a function!
The vertical line test shows that some x-values have two corresponding y-values.

STEP 11

This graph is a parabola, which is a quadratic relationship, not a linear one.

STEP 12

It is a function because it passes the vertical line test - each x-value has only one y-value.

STEP 13

The rate of change is the change in depth divided by the change in time.

STEP 14

For OA, Kat goes from 0m to -10m in 5 minutes.
So the rate is $\frac{-10 - 0}{5 - 0} = \frac{-10}{5} = -2$ m/min.

STEP 15

For AB, Kat stays at -10m for 5 minutes, so the rate is $\frac{-10 - (-10)}{10 - 5} = \frac{0}{5} = 0$ m/min.

STEP 16

For BC, Kat goes from -10m to -20m in 5 minutes.
The rate is $\frac{-20 - (-10)}{15 - 10} = \frac{-10}{5} = -2$ m/min.

STEP 17

For CD, Kat goes from -20m to -5m in 10 minutes.
The rate is $\frac{-5 - (-20)}{25 - 15} = \frac{15}{10} = 1.5$ m/min.

STEP 18

For DE, Kat goes from -5m to 0m in 5 minutes.
The rate is $\frac{0 - (-5)}{30 - 25} = \frac{5}{5} = 1$ m/min.

STEP 19

1. Campsite: Linear: yes, Function: yes
2. Fruit Flies: Linear: no, Function: yes
3. X-Graph: Linear: no, Function: no
4. Parabola: Linear: no, Function: yes
5. Kat's Dive: OA: -2 m/min, AB: 0 m/min, BC: -2 m/min, CD: 1.5 m/min, DE: 1 m/min

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Studdy Solution

STEP 1

STEP 2

1. Campsite Costs2. Fruit Fly Population3. X-Shaped Graph4. Parabolic Graph5. Kat's Dive

STEP 3

Let's **define** our variables! ccc is the total cost and ppp is the number of additional people.

STEP 4

The cost is $35\$35$35 plus $15\$15$15 per extra person.We can write this as c=35+15⋅pc = 35 + 15 \cdot pc=35+15⋅p.

STEP 5

This is a **linear relationship** because it's in the form y=mx+by = mx + by=mx+b, where m=15m = 15m=15 and b=35b = 35b=35.It's also a **function** because each input (ppp) has exactly one output (ccc).

STEP 6

Let PPP be the population and ttt be the time in hours.

STEP 7

Doubling every 12 hours means we multiply by 2 every 12 hours.This can be written as P(t)=P0⋅2t12P(t) = P_0 \cdot 2^{\frac{t}{12}}P(t)=P0​⋅212t​, where P0P_0P0​ is the **initial population**.

STEP 8

This is *not* linear because of the exponent.It's an **exponential relationship**.It *is* a **function** because each time has one population.

STEP 9

This graph is made of two intersecting lines, so it's *not* a single linear relationship.

STEP 10

It's *not* a function!The **vertical line test** shows that some x-values have *two* corresponding y-values.

STEP 11

This graph is a parabola, which is a quadratic relationship, not a linear one.

STEP 12

It *is* a **function** because it passes the **vertical line test** - each x-value has only one y-value.

STEP 13

The **rate of change** is the change in depth divided by the change in time.

STEP 14

For **OA**, Kat goes from 0m to -10m in 5 minutes.So the rate is −10−05−0=−105=−2\frac{-10 - 0}{5 - 0} = \frac{-10}{5} = -25−0−10−0​=5−10​=−2 m/min.

STEP 15

For **AB**, Kat stays at -10m for 5 minutes, so the rate is −10−(−10)10−5=05=0\frac{-10 - (-10)}{10 - 5} = \frac{0}{5} = 010−5−10−(−10)​=50​=0 m/min.

STEP 16

For **BC**, Kat goes from -10m to -20m in 5 minutes.The rate is −20−(−10)15−10=−105=−2\frac{-20 - (-10)}{15 - 10} = \frac{-10}{5} = -215−10−20−(−10)​=5−10​=−2 m/min.

STEP 17

For **CD**, Kat goes from -20m to -5m in 10 minutes.The rate is −5−(−20)25−15=1510=1.5\frac{-5 - (-20)}{25 - 15} = \frac{15}{10} = 1.525−15−5−(−20)​=1015​=1.5 m/min.

STEP 18

For **DE**, Kat goes from -5m to 0m in 5 minutes.The rate is 0−(−5)30−25=55=1\frac{0 - (-5)}{30 - 25} = \frac{5}{5} = 130−250−(−5)​=55​=1 m/min.

STEP 19

1. Campsite: **Linear: yes, Function: yes**2. Fruit Flies: **Linear: no, Function: yes**3. X-Graph: **Linear: no, Function: no**4. Parabola: **Linear: no, Function: yes**5. Kat's Dive: **OA: -2 m/min, AB: 0 m/min, BC: -2 m/min, CD: 1.5 m/min, DE: 1 m/min**

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1. Campsite Costs
2. Fruit Fly Population
3. X-Shaped Graph
4. Parabolic Graph
5. Kat's Dive

Let's define our variables! $c$ is the total cost and $p$ is the number of additional people.

The cost is $\$35$ plus $\$15$ per extra person.
We can write this as $c = 35 + 15 \cdot p$ .

This is a linear relationship because it's in the form $y = mx + b$ , where $m = 15$ and $b = 35$ .
It's also a function because each input ( $p$ ) has exactly one output ( $c$ ).

Let $P$ be the population and $t$ be the time in hours.

Doubling every 12 hours means we multiply by 2 every 12 hours.
This can be written as $P(t) = P_0 \cdot 2^{\frac{t}{12}}$ , where $P_0$ is the initial population.

This is not linear because of the exponent.
It's an exponential relationship.
It is a function because each time has one population.

This graph is made of two intersecting lines, so it's not a single linear relationship.

It's not a function!
The vertical line test shows that some x-values have two corresponding y-values.

It is a function because it passes the vertical line test - each x-value has only one y-value.

The rate of change is the change in depth divided by the change in time.

For OA, Kat goes from 0m to -10m in 5 minutes.
So the rate is $\frac{-10 - 0}{5 - 0} = \frac{-10}{5} = -2$ m/min.

For AB, Kat stays at -10m for 5 minutes, so the rate is $\frac{-10 - (-10)}{10 - 5} = \frac{0}{5} = 0$ m/min.

For BC, Kat goes from -10m to -20m in 5 minutes.
The rate is $\frac{-20 - (-10)}{15 - 10} = \frac{-10}{5} = -2$ m/min.

For CD, Kat goes from -20m to -5m in 10 minutes.
The rate is $\frac{-5 - (-20)}{25 - 15} = \frac{15}{10} = 1.5$ m/min.

For DE, Kat goes from -5m to 0m in 5 minutes.
The rate is $\frac{0 - (-5)}{30 - 25} = \frac{5}{5} = 1$ m/min.

1. Campsite: Linear: yes, Function: yes
2. Fruit Flies: Linear: no, Function: yes
3. X-Graph: Linear: no, Function: no
4. Parabola: Linear: no, Function: yes
5. Kat's Dive: OA: -2 m/min, AB: 0 m/min, BC: -2 m/min, CD: 1.5 m/min, DE: 1 m/min