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: $$\text{Misplaced hline}$$

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?