Solved on Dec 08, 2023

Find the point that satisfies the equation $y=8\left(\frac{1}{4}\right)^{x}$ . Options: $(-2,0)$ , $(-1,-2)$ , $(1,2)$ , $(2,1)$ .

STEP 1

Assumptions
1. The function is given by $y = 8\left(\frac{1}{4}\right)^{x}$ .
2. We need to determine which of the given points is a solution to the function.
3. A point $(x, y)$ is a solution to the function if, when $x$ is substituted into the function, the result is $y$ .

STEP 2

We will substitute each $x$ -coordinate from the given points into the function and check if the corresponding $y$ -coordinate is the result.

STEP 3

Start with the point $(-2,0)$ . Substitute $x = -2$ into the function and see if the resulting $y$ is 0.
$y = 8\left(\frac{1}{4}\right)^{-2}$

STEP 4

Calculate the value of $\left(\frac{1}{4}\right)^{-2}$ .
$\left(\frac{1}{4}\right)^{-2} = \left(4\right)^{2}$

STEP 5

Now calculate $4^2$ .
$4^2 = 16$

STEP 6

Multiply the result by 8.
$y = 8 \times 16$

STEP 7

Calculate the value of $y$ .
$y = 8 \times 16 = 128$

STEP 8

Since $y = 128$ and not 0, the point $(-2,0)$ is not a solution to the function.

STEP 9

Now, consider the point $(-1,-2)$ . Substitute $x = -1$ into the function and see if the resulting $y$ is -2.
$y = 8\left(\frac{1}{4}\right)^{-1}$

STEP 10

Calculate the value of $\left(\frac{1}{4}\right)^{-1}$ .
$\left(\frac{1}{4}\right)^{-1} = 4$

STEP 11

Multiply the result by 8.
$y = 8 \times 4$

STEP 12

Calculate the value of $y$ .
$y = 8 \times 4 = 32$

STEP 13

Since $y = 32$ and not -2, the point $(-1,-2)$ is not a solution to the function.

STEP 14

Now, consider the point $(1,2)$ . Substitute $x = 1$ into the function and see if the resulting $y$ is 2.
$y = 8\left(\frac{1}{4}\right)^{1}$

STEP 15

Calculate the value of $\left(\frac{1}{4}\right)^{1}$ .
$\left(\frac{1}{4}\right)^{1} = \frac{1}{4}$

STEP 16

Multiply the result by 8.
$y = 8 \times \frac{1}{4}$

STEP 17

Calculate the value of $y$ .
$y = 8 \times \frac{1}{4} = 2$

STEP 18

Since $y = 2$ , the point $(1,2)$ is a solution to the function.

STEP 19

For completeness, consider the point $(2,1)$ . Substitute $x = 2$ into the function and see if the resulting $y$ is 1.
$y = 8\left(\frac{1}{4}\right)^{2}$

STEP 20

Calculate the value of $\left(\frac{1}{4}\right)^{2}$ .
$\left(\frac{1}{4}\right)^{2} = \frac{1}{16}$

STEP 21

Multiply the result by 8.
$y = 8 \times \frac{1}{16}$

STEP 22

Calculate the value of $y$ .
$y = 8 \times \frac{1}{16} = \frac{1}{2}$

STEP 23

Since $y = \frac{1}{2}$ and not 1, the point $(2,1)$ is not a solution to the function.

STEP 24

The only point that resulted in a correct $y$ value when substituted into the function was $(1,2)$ .
The point that represents a solution of the function is $(1,2)$ .

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Find the point that satisfies the equation y=8(14)xy=8\left(\frac{1}{4}\right)^{x}y=8(41​)x. Options: (−2,0)(-2,0)(−2,0), (−1,−2)(-1,-2)(−1,−2), (1,2)(1,2)(1,2), (2,1)(2,1)(2,1).

STEP 1

Assumptions1. The function is given by y=8(14)x y = 8\left(\frac{1}{4}\right)^{x} y=8(41​)x.2. We need to determine which of the given points is a solution to the function.3. A point (x,y)(x, y)(x,y) is a solution to the function if, when xxx is substituted into the function, the result is yyy.

STEP 2

We will substitute each xxx-coordinate from the given points into the function and check if the corresponding yyy-coordinate is the result.

STEP 3

Start with the point (−2,0)(-2,0)(−2,0). Substitute x=−2x = -2x=−2 into the function and see if the resulting yyy is 0.y=8(14)−2y = 8\left(\frac{1}{4}\right)^{-2}y=8(41​)−2

STEP 4

Calculate the value of (14)−2\left(\frac{1}{4}\right)^{-2}(41​)−2.(14)−2=(4)2\left(\frac{1}{4}\right)^{-2} = \left(4\right)^{2}(41​)−2=(4)2

STEP 5

Now calculate 424^242.42=164^2 = 1642=16

STEP 6

Multiply the result by 8.y=8×16y = 8 \times 16y=8×16

STEP 7

Calculate the value of yyy.y=8×16=128y = 8 \times 16 = 128y=8×16=128

STEP 8

Since y=128y = 128y=128 and not 0, the point (−2,0)(-2,0)(−2,0) is not a solution to the function.

STEP 9

Now, consider the point (−1,−2)(-1,-2)(−1,−2). Substitute x=−1x = -1x=−1 into the function and see if the resulting yyy is -2.y=8(14)−1y = 8\left(\frac{1}{4}\right)^{-1}y=8(41​)−1

STEP 10

Calculate the value of (14)−1\left(\frac{1}{4}\right)^{-1}(41​)−1.(14)−1=4\left(\frac{1}{4}\right)^{-1} = 4(41​)−1=4

STEP 11

Multiply the result by 8.y=8×4y = 8 \times 4y=8×4

STEP 12

Calculate the value of yyy.y=8×4=32y = 8 \times 4 = 32y=8×4=32

STEP 13

Since y=32y = 32y=32 and not -2, the point (−1,−2)(-1,-2)(−1,−2) is not a solution to the function.

STEP 14

Now, consider the point (1,2)(1,2)(1,2). Substitute x=1x = 1x=1 into the function and see if the resulting yyy is 2.y=8(14)1y = 8\left(\frac{1}{4}\right)^{1}y=8(41​)1

STEP 15

Calculate the value of (14)1\left(\frac{1}{4}\right)^{1}(41​)1.(14)1=14\left(\frac{1}{4}\right)^{1} = \frac{1}{4}(41​)1=41​

STEP 16

Multiply the result by 8.y=8×14y = 8 \times \frac{1}{4}y=8×41​

STEP 17

Calculate the value of yyy.y=8×14=2y = 8 \times \frac{1}{4} = 2y=8×41​=2

STEP 18

Since y=2y = 2y=2, the point (1,2)(1,2)(1,2) is a solution to the function.

STEP 19

For completeness, consider the point (2,1)(2,1)(2,1). Substitute x=2x = 2x=2 into the function and see if the resulting yyy is 1.y=8(14)2y = 8\left(\frac{1}{4}\right)^{2}y=8(41​)2

STEP 20

Calculate the value of (14)2\left(\frac{1}{4}\right)^{2}(41​)2.(14)2=116\left(\frac{1}{4}\right)^{2} = \frac{1}{16}(41​)2=161​

STEP 21

Multiply the result by 8.y=8×116y = 8 \times \frac{1}{16}y=8×161​

STEP 22

Calculate the value of yyy.y=8×116=12y = 8 \times \frac{1}{16} = \frac{1}{2}y=8×161​=21​

STEP 23

Since y=12y = \frac{1}{2}y=21​ and not 1, the point (2,1)(2,1)(2,1) is not a solution to the function.

STEP 24

The only point that resulted in a correct yyy value when substituted into the function was (1,2)(1,2)(1,2).The point that represents a solution of the function is (1,2)(1,2)(1,2).

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Find the point that satisfies the equation $y=8\left(\frac{1}{4}\right)^{x}$ . Options: $(-2,0)$ , $(-1,-2)$ , $(1,2)$ , $(2,1)$ .

Assumptions
1. The function is given by $y = 8\left(\frac{1}{4}\right)^{x}$ .
2. We need to determine which of the given points is a solution to the function.
3. A point $(x, y)$ is a solution to the function if, when $x$ is substituted into the function, the result is $y$ .

We will substitute each $x$ -coordinate from the given points into the function and check if the corresponding $y$ -coordinate is the result.

Start with the point $(-2,0)$ . Substitute $x = -2$ into the function and see if the resulting $y$ is 0.
$y = 8\left(\frac{1}{4}\right)^{-2}$

Calculate the value of $\left(\frac{1}{4}\right)^{-2}$ .
$\left(\frac{1}{4}\right)^{-2} = \left(4\right)^{2}$

Now calculate $4^2$ .
$4^2 = 16$

Multiply the result by 8.
$y = 8 \times 16$

Calculate the value of $y$ .
$y = 8 \times 16 = 128$

Since $y = 128$ and not 0, the point $(-2,0)$ is not a solution to the function.

Now, consider the point $(-1,-2)$ . Substitute $x = -1$ into the function and see if the resulting $y$ is -2.
$y = 8\left(\frac{1}{4}\right)^{-1}$

Calculate the value of $\left(\frac{1}{4}\right)^{-1}$ .
$\left(\frac{1}{4}\right)^{-1} = 4$

Multiply the result by 8.
$y = 8 \times 4$

Calculate the value of $y$ .
$y = 8 \times 4 = 32$

Since $y = 32$ and not -2, the point $(-1,-2)$ is not a solution to the function.

Now, consider the point $(1,2)$ . Substitute $x = 1$ into the function and see if the resulting $y$ is 2.
$y = 8\left(\frac{1}{4}\right)^{1}$

Calculate the value of $\left(\frac{1}{4}\right)^{1}$ .
$\left(\frac{1}{4}\right)^{1} = \frac{1}{4}$

Multiply the result by 8.
$y = 8 \times \frac{1}{4}$

Calculate the value of $y$ .
$y = 8 \times \frac{1}{4} = 2$

Since $y = 2$ , the point $(1,2)$ is a solution to the function.

For completeness, consider the point $(2,1)$ . Substitute $x = 2$ into the function and see if the resulting $y$ is 1.
$y = 8\left(\frac{1}{4}\right)^{2}$

Calculate the value of $\left(\frac{1}{4}\right)^{2}$ .
$\left(\frac{1}{4}\right)^{2} = \frac{1}{16}$

Multiply the result by 8.
$y = 8 \times \frac{1}{16}$

Calculate the value of $y$ .
$y = 8 \times \frac{1}{16} = \frac{1}{2}$

Since $y = \frac{1}{2}$ and not 1, the point $(2,1)$ is not a solution to the function.

The only point that resulted in a correct $y$ value when substituted into the function was $(1,2)$ .
The point that represents a solution of the function is $(1,2)$ .