Math / Algebra

QuestionQuestion 6 Here is the graph of an exponential function $f$ . Find an equation defining $f$ . $f(x)=$ $\square$ type your answer...

Studdy Solution

STEP 1

1. The function $f(x)$ is an exponential function.
2. The graph passes through the points $(0, 100)$ , $(1, 200)$ , and $(2, 400)$ .
3. We need to find the equation of the function $f(x)$ .

STEP 2

1. Identify the general form of an exponential function.
2. Use the given points to determine the parameters of the function.
3. Write the equation of the function.

STEP 3

The general form of an exponential function is $f(x) = a \cdot b^x$ , where $a$ is the initial value (when $x = 0$ ) and $b$ is the base or growth factor.

STEP 4

Since the function passes through the point $(0, 100)$ , we can substitute into the general form to find $a$ : $f(0) = a \cdot b^0 = a = 100$

STEP 5

Use the point $(1, 200)$ to find $b$ : $f(1) = 100 \cdot b^1 = 200$ $100b = 200$ $b = \frac{200}{100} = 2$

STEP 6

Now that we have $a = 100$ and $b = 2$ , the equation of the function is: $f(x) = 100 \cdot 2^x$
The equation defining the exponential function $f$ is:
$f(x) = 100 \cdot 2^x$

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QuestionQuestion 6 Here is the graph of an exponential function $f$ . Find an equation defining $f$ . $f(x)=$ $\square$ type your answer...

Studdy Solution

STEP 1

1. The function $f(x)$ is an exponential function.
2. The graph passes through the points $(0, 100)$ , $(1, 200)$ , and $(2, 400)$ .
3. We need to find the equation of the function $f(x)$ .

STEP 2

1. Identify the general form of an exponential function.
2. Use the given points to determine the parameters of the function.
3. Write the equation of the function.

STEP 3

The general form of an exponential function is $f(x) = a \cdot b^x$ , where $a$ is the initial value (when $x = 0$ ) and $b$ is the base or growth factor.

STEP 4

Since the function passes through the point $(0, 100)$ , we can substitute into the general form to find $a$ : $f(0) = a \cdot b^0 = a = 100$

STEP 5

Use the point $(1, 200)$ to find $b$ : $f(1) = 100 \cdot b^1 = 200$ $100b = 200$ $b = \frac{200}{100} = 2$

STEP 6

Now that we have $a = 100$ and $b = 2$ , the equation of the function is: $f(x) = 100 \cdot 2^x$
The equation defining the exponential function $f$ is:
$f(x) = 100 \cdot 2^x$

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QuestionQuestion 6 Here is the graph of an exponential function fff. Find an equation defining fff. f(x)=f(x)=f(x)= □\square□ type your answer...

Studdy Solution

STEP 1

1. The function f(x) f(x) f(x) is an exponential function.2. The graph passes through the points (0,100) (0, 100) (0,100), (1,200) (1, 200) (1,200), and (2,400) (2, 400) (2,400).3. We need to find the equation of the function f(x) f(x) f(x).

STEP 2

1. Identify the general form of an exponential function.2. Use the given points to determine the parameters of the function.3. Write the equation of the function.

STEP 3

The general form of an exponential function is f(x)=a⋅bx f(x) = a \cdot b^x f(x)=a⋅bx, where a a a is the initial value (when x=0 x = 0 x=0) and b b b is the base or growth factor.

STEP 4

Since the function passes through the point (0,100) (0, 100) (0,100), we can substitute into the general form to find a a a: f(0)=a⋅b0=a=100 f(0) = a \cdot b^0 = a = 100 f(0)=a⋅b0=a=100

STEP 5

Use the point (1,200) (1, 200) (1,200) to find b b b: f(1)=100⋅b1=200 f(1) = 100 \cdot b^1 = 200 f(1)=100⋅b1=200 100b=200 100b = 200 100b=200 b=200100=2 b = \frac{200}{100} = 2 b=100200​=2

STEP 6

Now that we have a=100 a = 100 a=100 and b=2 b = 2 b=2, the equation of the function is: f(x)=100⋅2x f(x) = 100 \cdot 2^x f(x)=100⋅2x The equation defining the exponential function f f f is:f(x)=100⋅2x f(x) = 100 \cdot 2^x f(x)=100⋅2x

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QuestionQuestion 6 Here is the graph of an exponential function $f$ . Find an equation defining $f$ . $f(x)=$ $\square$ type your answer...

1. The function $f(x)$ is an exponential function.
2. The graph passes through the points $(0, 100)$ , $(1, 200)$ , and $(2, 400)$ .
3. We need to find the equation of the function $f(x)$ .

1. Identify the general form of an exponential function.
2. Use the given points to determine the parameters of the function.
3. Write the equation of the function.

The general form of an exponential function is $f(x) = a \cdot b^x$ , where $a$ is the initial value (when $x = 0$ ) and $b$ is the base or growth factor.

Since the function passes through the point $(0, 100)$ , we can substitute into the general form to find $a$ : $f(0) = a \cdot b^0 = a = 100$

Use the point $(1, 200)$ to find $b$ : $f(1) = 100 \cdot b^1 = 200$ $100b = 200$ $b = \frac{200}{100} = 2$

Now that we have $a = 100$ and $b = 2$ , the equation of the function is: $f(x) = 100 \cdot 2^x$
The equation defining the exponential function $f$ is:
$f(x) = 100 \cdot 2^x$