Question

NO LINKS!!! URGENT HELP PLEASE!!!!

Please help me with #4 - 6

For each table, state if the model is linear or exponential and write an equation

Answer 1

`\textsf{4)} \quad \textsf{Exponential:} \quad y=192 \cdot 4^x`

`\textsf{5)} \quad \textsf{Exponential:} \quad y=8\cdot \left((1)/(2)\right)^x`

`\textsf{6)} \quad \textsf{Linear:} \quad y=-2.5x-37`

Explanation:

Linear function

When the x-values increase by a constant amount, the y-values have a constant difference.

`\boxed{\begin{minipage}{6.3 cm}\underline{Linear Function}\\\\$f(x)=ax+b$\\\\where:\\ \phantom{ww}$\bullet$ $a$ is the slope. \\ \phantom{ww}$\bullet$ $b$ is the $y$-intercept.\\\end{minipage}}`

Exponential function

When the x-values increase by a constant amount, the y-values have a constant ratio.

`\boxed{\begin{minipage}{9 cm}\underline{Exponential Function}\\\\$f(x)=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}`

Question 4

From inspection of the given table, as the x-values increase by one, the y-values are 4 times the previous y-value. Therefore, they have a constant ratio of 4. This means that the equation is exponential.

The initial value "a" is the y-intercept, so a = 192.

The y-values have a growth factor of 4, so b = 4.

Substitute these values into the formula to create an equation for the table of values:

`\boxed{y=192 \cdot 4^x}`

Question 5

From inspection of the given table, as the x-values increase by one, the y-values are half the previous y-value. Therefore, they have a constant ratio of 1/2. This means that the equation is exponential.

The initial value "a" is the y-intercept, so a = 8.

The y-values have a growth factor of 1/2, so b = 1/2.

Substitute these values into the formula to create an equation for the table of values:

`\boxed{y=8\cdot \left((1)/(2)\right)^x}`

Question 6

From inspection of the given table, as the x-values increase by one, the y-values decrease by 2.5. Therefore, they have a constant difference of -2.5. This means that the equation is linear.

The slope "a" is the change in y-values divided by the change in x-values. As the y-values decrease by 2.5 for ever increase in one in the x-values, a = -2.5.

The y-intercept is the y-value when x = 0, so b = -37.

Substitute these values into the formula to create an equation for the table of values:

`\boxed{y=-2.5x-37}`

Answer 2

4. y = 192 * 4^x

5. y=8*2^(-x)

6. y = -2.5x - 37

Explanation:

4.

From the given data, we can see that as x increases by 1, y increases by a factor of 4.

Using the second and third data points, we can find a and b as:

When x = -2, y = 12, so we have:

12 = ab^(-2)

When x = -1, y = 48, so we have:

48 = ab^(-1)

Dividing the second equation by the first, we get:

4 = b^1

So b = 4, and substituting this into the first equation, we get:

12 = a(4)^(-2)

Simplifying, we get:

a = 192

So the equation for this exponential relationship is:

y = 192(4)^x

Simplifying further:

y = 192 * 4^x

5.

From the given data, we can see that as x increases by 1, y decreases by a factor of 2.

Using the second and third data points, we can find a and b as:

When x = -2, y = 32, so we have:

32 = ab^(-2)

When x = -1, y = 16, so we have:

16 = ab^(-1)

Dividing the second equation by the first, we get:

16/32=b^(-2+1)

1/2=b^(-1)

b=2

So b = 2, and substituting this into the first equation, we get:

32 = a*2^(-2)

32= 4a

a=32/4

a=8

So the equation for this exponential relationship is:

y = 8(2)^(-x)

Simplifying further:

y=8*2^(-x)

6.

The data suggests that as x increases by 1, y decreases by a constant amount. This suggests that the relationship between x and y is linear.

To find the equation for this relationship, we can use the slope-intercept form of a linear equation:

y = mx + b

where m is the slope and b is the y-intercept.

To find the values of m and b, we can use any two data points. Let's use the first and last data points:

When x = -3, y = -29.5, so we have:

-29.5 = m(-3) + b

When x = 3, y = -44.5, so we have:

-44.5 = m(3) + b

We can now solve for m and b. Subtracting the first equation from the second equation, we get:

-15 = 6m

So, m = -2.5.

Substituting this value into the first equation, we get:

-29.5 = (-2.5)(-3) + b

-29.5=7.5+b

b=-29.5-7.5

So, b =-37

Therefore, the equation for this linear relationship is:

y = -2.5x - 37