Question

Suppose that the water level of a river is 45 feet after 3 days of no rain and 10 feet after 10 days of no rain. Given the two points we know: (3, 45) and (10, 10), what would the point slope equation look like?(1 point)
Responses

(y−45)=−5(x−3)
open paren y minus 45 close paren is equal to negative 5 times open paren x minus 3 close paren

(y−10)=−5(x+10)
open paren y minus 10 close paren is equal to negative 5 times open paren x plus 10 close paren

(y+10)=−5(x−10)
open paren y plus 10 close paren is equal to negative 5 times open paren x minus 10 close paren

(y−3)=−5(x−45)

Answer 1

Function A: (four points on a graph) (-4, 0) (-2, 1) (0, 2) (2, 3)
Function B: (table chart)
x y
0 3
4 -9
8 -21
Function C: y=x+5
(4 points)
a. Which function has the greatest rate of change (slope)?

b. Which function has the smallest y-intercept?

c. Which function would you want to represent your money earned per hour?

d. Which function would give you the highest initial amount of money?

responses
function A
function B
function C

Answer 2

To find the point-slope equation from the two points we have: (3, 45) and (10, 10), we first need to find the slope of the line (m) between these two points.

The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points (3, 45) (i.e., \((x_1, y_1) = (3, 45)\)) and (10, 10) (i.e., \((x_2, y_2) = (10, 10)\)), we can substitute these values in:

\[
m = \frac{10 - 45}{10 - 3} = \frac{-35}{7} = -5
\]

Now that we have the slope (m = -5), we can use the point-slope form of the equation of a line, which is:

\[
y - y_1 = m(x - x_1)
\]

Choosing point (3, 45), we substitute into the equation:

\[
y - 45 = -5(x - 3)
\]

So the point-slope equation that corresponds to the given points is:

\((y−45) = −5(x−3)\)

Therefore, the correct response is:

**(y−45)=−5(x−3)**

Answer 3

To analyze the functions provided, let's break each question down step by step:

### Function A
Points: (-4, 0), (-2, 1), (0, 2), (2, 3)

**To find the slope (rate of change):**
Using two points, for example, (0, 2) and (2, 3):
\[
m = \frac{3 - 2}{2 - 0} = \frac{1}{2}
\]
You can calculate the slope between any two points to find this was consistent; the slope is \( \frac{1}{2} \).

### Function B
Table:
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 3 \\
4 & -9 \\
8 & -21 \\
\hline
\end{array}
\]

**Finding the slope between the first two points (0, 3) and (4, -9):**
\[
m = \frac{-9 - 3}{4 - 0} = \frac{-12}{4} = -3
\]

**Finding the slope between the second two points (4, -9) and (8, -21):**
\[
m = \frac{-21 - (-9)}{8 - 4} = \frac{-12}{4} = -3
\]
The slope is consistently \(-3\).

### Function C
Equation: \(y = x + 5\)

**The slope (rate of change) is:**
1 (the coefficient of x).

### Comparing Slopes
- **Function A:** slope = \( \frac{1}{2} \)
- **Function B:** slope = \(-3\)
- **Function C:** slope = \(1\)

**Greatest Rate of Change:**
- Function C has the greatest slope of \(1\).

### Finding y-intercepts
- **Function A's y-intercept:** When \(x = 0\), \(y = 2\).
- **Function B's y-intercept:** When \(x = 0\), \(y = 3\).
- **Function C's y-intercept:** When \(x = 0\), \(y = 5\).

**Smallest Y-Intercept:**
- **Function A:** \(2\)
- **Function B:** \(3\)
- **Function C:** \(5\)

The smallest y-intercept is from **Function A**.

### Money Earned per Hour
Generally, you might prefer a function with a positive slope representing earnings, and you want it to be as high as possible.

- Function A has a positive slope (\(\frac{1}{2}\)).
- Function C has a slope of \(1\).

**Function C** would represent the best option for money earned per hour since it has a positive slope and does not dip.

### Highest Initial Amount of Money
The initial amount of money is represented by the y-intercept.
- Function A = \(2\)
- Function B = \(3\)
- Function C = \(5\)

The highest initial amount of money is from **Function C**.

### Answers
a. Function C (greatest rate of change)
b. Function A (smallest y-intercept)
c. Function C (best for money earned per hour)
d. Function C (highest initial amount of money)

Would you like to clarify further or provide more detail for any of these points?