Question

Write an equation for the linear function f with the given value: f(–3) = 3 and f(3) = –1.

Answer 1

Given:

f(-3) = 3

f(3) = -1

Let's write an equation for the linear function.

For example:

f(x) = y

We have the point:

(x, y)

From, f(-3) = 3, we have the point:

(-3, 3)

From f(3) = -1, we have the point:

(3, -1)

Thus, we have the points:

(x1, y1) ==> (-3, 3)

(x2, y2) ==> (3, -1)

To write a linear equation, applt the slope-intercept form:

y = mx + b

Where m is the slope and b is the y-intercept.

To find the slope, apply the formula:

`\begin{gathered} m=(y2-y1)/(x2-x1) \\ \\ m=(-1-3)/(3-(-3)) \\ \\ m=(-1-3)/(3+3) \\ \\ m=(-4)/(6) \\ \\ m=-(2)/(3) \end{gathered}`

The slope is -⅔

`y=-(2)/(3)x+b`

To solve for the y-intercept, b, susbstitute either of the points for the value of x and y.

Take the point: (-3, 3), substitute -3 for x and 3 for y:

`\begin{gathered} 3=-(2)/(3)\ast-3+b \\ \\ 3=2+b \\ \\ \text{Subtract 2 from both sides:} \\ 3-2=2-2+b \\ \\ 1=b \\ \\ b=1 \end{gathered}`

The y-intercept, b is = 1

Therefore, the equation for thr linear function is:

`y=-(2)/(3)x+1`

ANSWER:

`y=-(2)/(3)x+1`