Question

Let f be a linear function such that f(1) = 5 and f(3) = 9. Find an equation for f(x).

Answer 1

`f(x)=2x+3`

Explanation:

So we know that:

`f(1)=5 \text{ and } f(3)=9`

In other words, we have 2 coordinates: (1,5) and (3,9).

With this, we can figure out the slope. Let (1,5) be x₁ and y₁ and let (3,9) be x₂ and y₂. The slope formula is:

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

Substitute:

`m=(9-5)/(3-1)\\m=4/2=2`

So, the slope is 2.

Now, use the point-slope form to figure out the y-intercept. The point-slope form is:

`y-y_1=m(x-x_1)`

For consistency, let's continue to use (1,5) as x₁ and y₁. Now, we can use 2 as m. Thus:

`y-(5)=2(x-1)`

Distribute:

`y-5=2x-2`

Add 5 to both sides:

`y=2x+3`

So:

`f(x)=2x+3`

And that's our equation :)