Question

How do you find the gradient of a tangent? (full question below)

I already figured out the point of intersection but can't seem to figure out the gradient of the tangents?

Answer 1

The gradient when `y=x^2-9x` is `-7`
The gradient when `y=x^2-9` is `2`

Explanation:

Finding the point of intersection of two parabolas means that your two equations are equal at a point

`x^2-9=x^2-9x`
`-9=-9x`
`x=1`
When `x=1`, `y=-8` --> `(1,-8)`

To find the gradient of the lines at `(1,-8)`, we need to find the first derivative

`y=x^2-9`
`(dy)/(dx)=2x`
Therefore, when we sub `x=1`
`(dy)/(dx)=2`
The gradient when `y=x^2-9` is `2`

---

`y=x^2-9x`
`(dy)/(dx)=2x-9`
Therefore, when we sub `x=1`
`(dy)/(dx)=-7`
The gradient when `y=x^2-9x` is `-7`

graph{(y-x^2+9)(y-x^2+9x)=0 [-10, 10, -5, 5]}