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)

Question

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

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 2

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 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?