Question

NO LINKS!! URGENT HELP PLEASE!!!!

Please help me with #1 - 3


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

Answer 1

1. y = -5x + 15.

2. y = -x + 4.

3. y = 80(2^(x/3))

Step-by-step explanation:

1.

From the given data, we can see that as x increases by 1, y decreases by a fixed amount of 5. This means that the relationship between x and y is linear, specifically a decreasing linear relationship.

To write the equation for a linear relationship, we need to find the slope (m) and y-intercept (b).

Using the formula for slope:

m = (change in y) / (change in x)

m = (0 - 30) / (3 - (-3))

m = -5

Using the point-slope form of a linear equation:

y - y1 = m(x - x1)

y - 30 = -5(x - (-3))

y - 30 = -5x - 15

y = -5x + 15

So the equation for this linear relationship is y = -5x + 15.

2.

From the given data, we can see that as x increases by 3, y decreases by a fixed amount of 3. This means that the relationship between x and y is linear, specifically a decreasing linear relationship.

To write the equation for a linear relationship, we need to find the slope (m) and y-intercept (b).

Using the formula for slope:

m = (change in y) / (change in x)

m = (-5 - 13) / (9 - (-9))

m = -18 / 18

m = -1

Using the point-slope form of a linear equation:

y - y1 = m(x - x1)

y - 13 = -1(x - (-9))

y - 13 = -1(x + 9)

y = -x + 4

So the equation for this linear relationship is y = -x + 4.

3.

From the given data, we can see that as x increases by 1, y doubles. This means that the relationship between x and y is exponential.

To write the equation for an exponential relationship, we can use the general form of an exponential function:

y = ab^x

where a is the initial value, b is the base, and x is the exponent.

To find a and b, we can use the first and fourth data points since they have the smallest and largest values of y, respectively.

When x = -3, y = 10, so we have:

10 = ab^(-3)

When x = 0, y = 80, so we have:

80 = ab^(0)

From the second equation, we can see that a = 80. Substituting this into the first equation, we get:

10 = 80b^(-3)

Simplifying, we get:

b = 2^(1/3)

So the equation for this exponential relationship is:

y = 80(2^(1/3))^x

Simplifying further:

y = 80(2^(x/3))

Answer 2

Answers:

1. Linear; equation is y = -5x+15
2. Linear; equation is y = -x+4
3. Exponential; equation is y = 80*2^x

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Step-by-step explanation:

Problem 1

Each time x goes up by 1, y goes down by 5. This constant rate of change leads to the equation being linear. The slope is m = -5/1 = -5. You can use the slope formula for any two points in the table to confirm this is the correct slope.

The y intercept is b = 15 because we have x = 0 lead to y = 15.

Therefore, we go from y = mx+b to y = -5x+15

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Problem 2

Each time x goes up by 3, the y coordinate goes down by 3. Therefore, we have another linear equation here. The slope is m = -3/3 = -1.

The y intercept is b = 4 because x = 0 leads to y = 4.

y = mx+b turns into y = -x+4

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Problem 3

Each time x goes up by 1, y is NOT going up the same amount. The jump from 10 to 20 is +10. The jump from 20 to 40 is +20. And so on.

Therefore, this equation isn't linear. Instead it's exponential. Each y term doubles when x increases by 1, so that's why the b term is b = 2.

The initial term is a = 80 because x = 0 leads to y = 80.

We go from y = a*b^x to y = 80*2^x

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You can use a tool like Desmos or GeoGebra to visually confirm each of the answers. Both also offer support for function tables.