Question

The rate at which crickets chirp is a linear function of

temperature. At 60°F they chirp 74 times per minute.
At 72º they chirp 120 times per minute.
a.) Write an equation in slope intercept form that
represents this function.
b.)Predict the number of chirps per minute when the
temperature is 64º.

Answer 1

a)
`y=(23)/(6) x-156`

b)
`(268)/(3)`

Explanation:

Formulas for linear equations:

Slope-intercept:
`y=mx+b`, where m is the slope and b is the y-intercept

Point-slope:
`y-y_1=m(x-x_1)`, where (x₁, y₁) is a point on the function and m is the slope.

a) We are given two points. If you're given two points (x₁, y₁) and (x₂, y₂), the slope m is given by the formula:

`m = (y_2-y_1)/(x_2-x_1)`

Let's mark the temperature on the x-axis, and the number of chirps per minute on the y-axis. So, we have the following points: (60, 74), and (72, 120). Calculating the slope m between these two points:

`m=(120-74)/(72-60)`

`m=(23)/(6)`

Since we have the x and y coordinates of a point on the function, we can write an equation in point-slope form, which will help us to find the y-intercept. We need the y-intercept in order to write the slope intercept form of the equation. We will use the point (60, 74) here, although you can also use (72, 120) if you want.

Point-slope form equation of the linear function:

`y-74=(23)/(6) (x-60)`

Now that we have the point-slope equation of the function, we can find the y-intercept. The y-intercept is the point on the function where the graph crosses the y-axis (when the x-value is 0). So, we plug in 0 for x into the equation and solve for y:

`y-74=(23)/(6) (0-60)`

`y=-230+74`

`y=-156`

Now that we have the y-intercept value of the function, we can write an equation in slope-intercept form:

`y=(23)/(6) x-156`

b) Since we marked the temperature on the x-axis, we can just plug in the number 64 as the x-value into the equation we have. y will be the number of chirps per minute when the temperature is 64º.

`y=((23)/(6) *64)-156`

`y=(736)/(3) - (468)/(3)`

`y=(268)/(3)`

If you have any questions, leave a comment below.