Math / Algebra

QuestionFind the formula for an exponential function that passes through the two points given. $(0,5000)$ and $(3,5)$ $f(x)=$ $\square$ Question Help: Written Example Submit Question

Studdy Solution

STEP 1

What is this asking? We need to find the equation of an exponential function that goes through two specific points. Watch out! Remember that exponential functions have a specific format, and we need to carefully use the given points to find the correct formula.

STEP 2

1. Define the function
2. Use the first point
3. Use the second point
4. Write the final equation

STEP 3

Let's define our exponential function in its general form: $f(x) = a \cdot b^x$ Where $a$ is the initial value and $b$ is the base that determines how the function grows or decays.

STEP 4

We know that the function passes through the point $(0, 5000)$ .
This means when $x = 0$ , $f(x) = 5000$ .
Let's plug these values into our function: $5000 = a \cdot b^0$

STEP 5

Since any number (except 0) raised to the power of 0 is 1, we have $b^0 = 1$ .
This simplifies our equation: $5000 = a \cdot 1$ So, we've found our initial value: $a = 5000$ !

STEP 6

Now, let's use the second point $(3, 5)$ .
This tells us that when $x = 3$ , $f(x) = 5$ .
Let's plug these values, along with our newly discovered $a$ value, into our function: $5 = 5000 \cdot b^3$

STEP 7

To solve for $b$ , we first need to isolate it.
Let's divide both sides of the equation by 5000: $\frac{5}{5000} = \frac{5000 \cdot b^3}{5000}$ $\frac{1}{1000} = b^3$

STEP 8

Now, we'll take the cube root of both sides to find $b$ : $\sqrt[3]{\frac{1}{1000}} = \sqrt[3]{b^3}$ $\frac{\sqrt[3]{1}}{\sqrt[3]{1000}} = b$ $\frac{1}{10} = b$ So, $b = \frac{1}{10}$ !

STEP 9

With $a = 5000$ and $b = \frac{1}{10}$ , we can write the final equation: $f(x) = 5000 \cdot \left(\frac{1}{10}\right)^x$

STEP 10

$f(x) = 5000 \cdot \left(\frac{1}{10}\right)^x$

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QuestionFind the formula for an exponential function that passes through the two points given. $(0,5000)$ and $(3,5)$ $f(x)=$ $\square$ Question Help: Written Example Submit Question

Studdy Solution

STEP 1

What is this asking? We need to find the equation of an exponential function that goes through two specific points. Watch out! Remember that exponential functions have a specific format, and we need to carefully use the given points to find the correct formula.

STEP 2

1. Define the function
2. Use the first point
3. Use the second point
4. Write the final equation

STEP 3

Let's define our exponential function in its general form: $f(x) = a \cdot b^x$ Where $a$ is the initial value and $b$ is the base that determines how the function grows or decays.

STEP 4

We know that the function passes through the point $(0, 5000)$ .
This means when $x = 0$ , $f(x) = 5000$ .
Let's plug these values into our function: $5000 = a \cdot b^0$

STEP 5

Since any number (except 0) raised to the power of 0 is 1, we have $b^0 = 1$ .
This simplifies our equation: $5000 = a \cdot 1$ So, we've found our initial value: $a = 5000$ !

STEP 6

Now, let's use the second point $(3, 5)$ .
This tells us that when $x = 3$ , $f(x) = 5$ .
Let's plug these values, along with our newly discovered $a$ value, into our function: $5 = 5000 \cdot b^3$

STEP 7

To solve for $b$ , we first need to isolate it.
Let's divide both sides of the equation by 5000: $\frac{5}{5000} = \frac{5000 \cdot b^3}{5000}$ $\frac{1}{1000} = b^3$

STEP 8

Now, we'll take the cube root of both sides to find $b$ : $\sqrt[3]{\frac{1}{1000}} = \sqrt[3]{b^3}$ $\frac{\sqrt[3]{1}}{\sqrt[3]{1000}} = b$ $\frac{1}{10} = b$ So, $b = \frac{1}{10}$ !

STEP 9

With $a = 5000$ and $b = \frac{1}{10}$ , we can write the final equation: $f(x) = 5000 \cdot \left(\frac{1}{10}\right)^x$

STEP 10

$f(x) = 5000 \cdot \left(\frac{1}{10}\right)^x$

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QuestionFind the formula for an exponential function that passes through the two points given. (0,5000)(0,5000)(0,5000) and (3,5)(3,5)(3,5) f(x)=f(x)=f(x)= □\square□ Question Help: Written Example Submit Question

Studdy Solution

STEP 1

What is this asking? We need to find the equation of an exponential function that goes through two specific points. Watch out! Remember that exponential functions have a specific format, and we need to carefully use the given points to find the correct formula.

STEP 2

1. Define the function2. Use the first point3. Use the second point4. Write the final equation

STEP 3

Let's **define** our exponential function in its general form: f(x)=a⋅bx f(x) = a \cdot b^x f(x)=a⋅bx Where aaa is the **initial value** and bbb is the **base** that determines how the function grows or decays.

STEP 4

We know that the function passes through the point (0,5000)(0, 5000)(0,5000).This means when x=0x = 0x=0, f(x)=5000f(x) = 5000f(x)=5000.Let's **plug** these values into our function: 5000=a⋅b0 5000 = a \cdot b^0 5000=a⋅b0

STEP 5

Since *any* number (except 0) raised to the power of 0 is 1, we have b0=1b^0 = 1b0=1.This **simplifies** our equation: 5000=a⋅1 5000 = a \cdot 1 5000=a⋅1 So, we've **found** our initial value: a=5000a = 5000a=5000!

STEP 6

Now, let's **use** the second point (3,5)(3, 5)(3,5).This tells us that when x=3x = 3x=3, f(x)=5f(x) = 5f(x)=5.Let's **plug** these values, along with our newly discovered aaa value, into our function: 5=5000⋅b3 5 = 5000 \cdot b^3 5=5000⋅b3

STEP 7

To **solve** for bbb, we first need to **isolate** it.Let's **divide** both sides of the equation by 5000: 55000=5000⋅b35000 \frac{5}{5000} = \frac{5000 \cdot b^3}{5000} 50005​=50005000⋅b3​ 11000=b3 \frac{1}{1000} = b^3 10001​=b3

STEP 8

Now, we'll take the **cube root** of both sides to find bbb: 110003=b33 \sqrt[3]{\frac{1}{1000}} = \sqrt[3]{b^3} 310001​​=3b3​ 1310003=b \frac{\sqrt[3]{1}}{\sqrt[3]{1000}} = b 31000​31​​=b110=b \frac{1}{10} = b 101​=bSo, b=110b = \frac{1}{10}b=101​!

STEP 9

With a=5000a = 5000a=5000 and b=110b = \frac{1}{10}b=101​, we can **write** the final equation: f(x)=5000⋅(110)x f(x) = 5000 \cdot \left(\frac{1}{10}\right)^x f(x)=5000⋅(101​)x

STEP 10

f(x)=5000⋅(110)x f(x) = 5000 \cdot \left(\frac{1}{10}\right)^x f(x)=5000⋅(101​)x

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QuestionFind the formula for an exponential function that passes through the two points given. $(0,5000)$ and $(3,5)$ $f(x)=$ $\square$ Question Help: Written Example Submit Question

1. Define the function
2. Use the first point
3. Use the second point
4. Write the final equation

Let's define our exponential function in its general form: $f(x) = a \cdot b^x$ Where $a$ is the initial value and $b$ is the base that determines how the function grows or decays.

We know that the function passes through the point $(0, 5000)$ .
This means when $x = 0$ , $f(x) = 5000$ .
Let's plug these values into our function: $5000 = a \cdot b^0$

Since any number (except 0) raised to the power of 0 is 1, we have $b^0 = 1$ .
This simplifies our equation: $5000 = a \cdot 1$ So, we've found our initial value: $a = 5000$ !

Now, let's use the second point $(3, 5)$ .
This tells us that when $x = 3$ , $f(x) = 5$ .
Let's plug these values, along with our newly discovered $a$ value, into our function: $5 = 5000 \cdot b^3$

To solve for $b$ , we first need to isolate it.
Let's divide both sides of the equation by 5000: $\frac{5}{5000} = \frac{5000 \cdot b^3}{5000}$ $\frac{1}{1000} = b^3$

Now, we'll take the cube root of both sides to find $b$ : $\sqrt[3]{\frac{1}{1000}} = \sqrt[3]{b^3}$ $\frac{\sqrt[3]{1}}{\sqrt[3]{1000}} = b$ $\frac{1}{10} = b$ So, $b = \frac{1}{10}$ !

With $a = 5000$ and $b = \frac{1}{10}$ , we can write the final equation: $f(x) = 5000 \cdot \left(\frac{1}{10}\right)^x$

$f(x) = 5000 \cdot \left(\frac{1}{10}\right)^x$