Question

Write a linear equation that intersects ytte points. Then willes second linear equation that intersecta y at just one point, and ther equation that does not intersecty Expten how you found the linear equations 40

Answer 1

`(a)\ y = 2x +8`

`(b)\ y = 0`

`(c)\ y = -5`

Explanation:

Given

`y = x^2`

First, graph
`y = x^2` (See attachment)

We are not limited to any particular linear equation; as long as we meet the required condition

Solving (a): Equation of a line that intersects
`y = x^2` at two points

Select any two points on opposite sides of the curve

The selected points are:

`(x_1,y_1) = (-2,4)`

`(x_2,y_2) = (4,16)`

Calculate slope (m)

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

`m = (16 - 4)/(4 - -2)`

`m = (12)/(6)`

`m = 2`

The equation is then calculated using:

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

This gives:

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

`y = 2(x +2) + 4`

Open bracket

`y = 2x +4 + 4`

`y = 2x +8`

Solving (b): Equation of a line that intersects
`y = x^2` at one point

A good illustration of this is x-axis

Because it intersects with the line at just one point, which is origin (0,0).

The equation of line at this point is:

`y = 0`

Solving (c): Equation of a line that does not intersect
`y = x^2`

Here, we make use of any horizontal line below the x-axis

Select any two horizontal points below the x-axis

`(x_1,y_1) = (-5,-5)`

`(x_2,y_2) = (5,-5)`

Calculate slope

`m = (-5--5)/(5--5)`

`m = (0)/(10)`

`m = 0`

The equation is then calculated using:

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

`y = 0(x - -5) + (-5)`

`y = 0-5`

`y = -5`