Question

This graph shows a linear function whose domain is the set of integers.
the main points are (3,4)(6,3)
Write an equation for the function in the form f(x)=mx+b.
f(x)=

Answer 1

To find the equation of the linear function in the form \( f(x) = mx + b \), we first need to determine the slope \( m \) using the two given points: \( (3, 4) \) and \( (6, 3) \).

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

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

Plugging in the coordinates of the points:

- \( (x_1, y_1) = (3, 4) \)
- \( (x_2, y_2) = (6, 3) \)

\[
m = \frac{3 - 4}{6 - 3} = \frac{-1}{3} = -\frac{1}{3}
\]

Now we have the slope \( m = -\frac{1}{3} \).

Next, we use the point-slope form of a linear equation, which is:

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

We can use one of the points, for example \( (3, 4) \):

\[
y - 4 = -\frac{1}{3}(x - 3)
\]

Now we simplify this equation to the slope-intercept form \( y = mx + b \):

1. Distribute \( -\frac{1}{3} \):

\[
y - 4 = -\frac{1}{3}x + 1
\]

2. Add 4 to both sides:

\[
y = -\frac{1}{3}x + 1 + 4
\]
\[
y = -\frac{1}{3}x + 5
\]

Thus, the equation of the function is:

\[
f(x) = -\frac{1}{3}x + 5
\]