Question

name: kaliah salinasdate: feb 10 20267.3 exponential functions: characteristics & graph ws#11. bacteria in a culture grows exponentially with time. the number of bacteria cells quintuples (fivefold increase) from the previous daya) complete the table and graph|days|number of bacteria||----|----||0|12||1|60||2|300||3| |b) how many bacteria cells will there be on day 6?c) write an exponential function that models the number of bacteria cells in the format of $f(x)=a cdot b^{x}$.directions: a) complete the table, b) graph the function, c) state the y-intercept, and d) determine if function is exponential growth or decay.2. $f(x)=4^{x}$|$x$|$f(x)$||----|----||-2| ||-1| ||0| ||1| ||2| |c) y-intercept:d) growth or decay:3. $f(x)=3 cdot 2^{x}$|$x$|$f(x)$||----|----||-2| ||-1| ||0| ||1| ||2| |c) y-intercept:d) growth or decay:

Explanation:

Problem 1

Step1: Calculate day 3 bacteria count

Multiply day 2 count by 5:
$300 \times 5 = 1500$

Step2: Find day 6 bacteria count

Use exponential growth: $a=12$, $b=5$, $x=6$
$12 \times 5^6 = 12 \times 15625 = 187500$

Step3: Write exponential function

Identify $a$ (initial value) and $b$ (growth factor):
$f(x) = 12 \cdot 5^x$

Step4: Identify y-intercept

The y-intercept is the initial value at $x=0$:
$f(0) = 12 \cdot 5^0 = 12$

Step5: Classify growth/decay

Since $b=5>1$, it is exponential growth.

Step1: Calculate $f(x)$ values

Substitute $x$ into $f(x)=4^x$:

• $x=-2$: $4^{-2} = \frac{1}{4^2} = \frac{1}{16} = 0.0625$
• $x=-1$: $4^{-1} = \frac{1}{4} = 0.25$
• $x=0$: $4^0 = 1$
• $x=1$: $4^1 = 4$
• $x=2$: $4^2 = 16$

Step2: Identify y-intercept

The y-intercept is $f(0)=1$, so $(0,1)$

Step3: Classify growth/decay

Since $b=4>1$, it is exponential growth.

Step1: Calculate $f(x)$ values

Substitute $x$ into $f(x)=3 \cdot 2^x$:

• $x=-2$: $3 \cdot 2^{-2} = 3 \times \frac{1}{4} = \frac{3}{4} = 0.75$
• $x=-1$: $3 \cdot 2^{-1} = 3 \times \frac{1}{2} = \frac{3}{2} = 1.5$
• $x=0$: $3 \cdot 2^0 = 3 \times 1 = 3$
• $x=1$: $3 \cdot 2^1 = 3 \times 2 = 6$
• $x=2$: $3 \cdot 2^2 = 3 \times 4 = 12$

Step2: Identify y-intercept

The y-intercept is $f(0)=3$, so $(0,3)$

Step3: Classify growth/decay

Since $b=2>1$, it is exponential growth.

Answer:

a) Day 3 number of bacteria: 1500
b) Day 6 number of bacteria: 187500
c) $f(x) = 12 \cdot 5^x$
d) Exponential growth
y-intercept: $(0, 12)$

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Problem 2