Question

Use the table to answer the question.%0D%0A%0D%0Ax%0D%0A%0D%0Ay%0D%0A%0D%0A1%094%0D%0A−3%0D%0A−%0D%0A3%0D%0A8%0D%0AFind the rate of change and the initial value of the linear function given as a table of values.%0D%0A%0D%0A(1 point)%0D%0AThe rate of change is m=%0D%0A%0D%0A=%0D%0A, and the initial value is b=%0D%0A%0D%0A=%0D%0A.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the rate of change, we can use the formula:

\(\text{{rate of change}} = \frac{{\text{{change in y}}}}{{\text{{change in x}}}}\)

Using the table, we can see that when x changes from 1 to -3, y changes from 4 to 3.

So, the change in y is 3 - 4 = -1, and the change in x is -3 - 1 = -4.

Therefore, the rate of change is:

\(\text{{rate of change}} = \frac{{-1}}{{-4}} = \frac{1}{4}\)

To find the initial value, we can use the formula:

\(\text{{initial value}} = y - \text{{rate of change}} \cdot x\)

Using any of the given values in the table, we can substitute the values and solve for the initial value.

Let's use the first point (1, 4):

\(\text{{initial value}} = 4 - \frac{1}{4} \cdot 1 = 4 - \frac{1}{4} = \frac{15}{4}\)

So, the initial value is:

b = \(\frac{15}{4}\)

Therefore, the rate of change is m = \(\frac{1}{4}\), and the initial value is b = \(\frac{15}{4}\).

Answer 2

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Answer 3

[AI-generated answer — Bot GPT 3.5]

The rate of change is \(m = \frac{1}{4}\), and the initial value is \(b = \frac{15}{4}\).

Answer 4

[AI-generated answer — Explain Bot]

To find the rate of change and the initial value of the linear function given in the table, we can use the formula for the slope-intercept form of a linear equation, y = mx + b, where m represents the rate of change (slope) and b represents the initial value (y-intercept).

Looking at the table:

```
x y
1 4
−3
3 8
```

To find the rate of change (m), we can choose two points from the table and use the formula:

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

Let's choose the points (1, 4) and (3, 8) to calculate the rate of change:

(change in y) = 8 - 4 = 4
(change in x) = 3 - 1 = 2

Now we can substitute these values into the formula:

m = 4/2 = 2

Therefore, the rate of change (m) is 2.

To find the initial value (b), we can pick any point from the table and substitute its coordinates into the equation y = mx + b. Let's choose the point (1, 4):

4 = 2(1) + b

Simplifying the equation:

4 = 2 + b

To isolate b, we subtract 2 from both sides:

b = 4 - 2 = 2

Therefore, the initial value (b) is 2.

In summary, the rate of change (m) is 2, and the initial value (b) is 2.