Question

Organize the functions from the least to greatest according to their y-intercepts.?

Answer 1

`color(blue)(g(x),f(x),h(x)`

Explanation:

First `g(x)`

We have slope 4 and point `(2,3)`

Using point slope form of a line:

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

`y-3=4(x-2)`

`y=4x-5`

`g(x)=4x-5`

Intercept is `-5`

`f(x)`

From the graph you can see the y intercept is `-1`

`h(x)`:

Assuming these are all linear functions:

Using slope intercept form:

`y=mx+b`

Using first two rows of table:

`4=m(2)+b \ \ \[1]`

`5=m(4)+b \ \ \[2]`

Solving `[1]` and `[2]` simultaneously:

Subtract `[1]` from `[2]`

`1=2m=>m=1/2`

Substituting in `[1]`:

`4=1/2(2)+b=>b=3`

Equation:

`y=1/2x+3`

`h(x)=1/2x+3`

This has a y intercept of `3`

So from lowest intercept to highest:

`g(x),f(x),h(x)`

Answer 2

same as displayed

Explanation:

the equations for all linear functions can be arranged into the form `y = mx + c`, where

`m` is the slope (gradient - how steep the graph is)
`c` is the `y`-intercept (the `y`-value when `x = 0`)

'a function `g` has a slope of `4` and passes through the point `(2,3)`'.

we know that `m = 4`, and that when `x = 2`, `y = 3`.

since `y = mx + c`, we know that for this function `g`, `3 = (4*2) + c`

`3 = 8 + c`

`c = 3 - 8`

`c = -5`

hence, `c` (the `y`-intercept) is `-5` for the graph of `g(x)`..

-

next shown is the graph of `f(x)`.

the `y`-intercept can be seen here, as the `y`-value at the point where the graph meets the `y`-axis.

reading off the scale for the `y`-axis (`1` per square), you can see that `y = -2` when the graph meets the `y`-axis.

hence, `c = -2` for the graph of `f(x)`.

-

the table of values for the function `h(x)` give the `y`-values at `x = 2, x= 4` and `x = 6`.

we see that for each time `x` increases by `2`, `h(x)` or `y` increases by `1`.

this is the same pattern for decrease.
since `x = 0` is a decrease of `2` from `x =2`, we know that the value of `y` at `x = 0` is `1` less than `y`'s value at `x = 2`.

the `y`-value at `x = 2` is shown to be `4`.

`4 - 1 = 3`

when `x = 0`, `h(x) = 3`, and `y = 3`.

hence, `c = 3` for the graph of `h(x)`.

-

so we have

`c = -5` for `g(x)`
`c = -2` for `f(x)`
`c = 3` for `h(x)`

these are in order from smallest to largest, so the sequence should be the same as in the pictures.