Question

Helpppp fastttt Plz show allll workkkkk​

Answer 1

Hopefully that helps

Answer 2

`m=8x-5-4\Delta x`

Explanation:

We have the function:

`f(x)=4x^2-5x-1`

And we want to use the difference quotient:

`m=(f(x+\Delta x)-f(x))/(\Delta x)`

To find the slope function.

So, let's substitute
`x+\Delta x` for our function. This yields:

`m=((4(x+\Delta x)^2-5(x+\Delta x)-1)-(4x^2-5x-1)))/(\Delta x)`

Let's square. Use the perfect square trinomial pattern. This yields:

`m=((4(x^2+2x\Delta x-\Delta x^2)-5(x+\Delta x)-1)-(4x^2-5x-1)))/(\Delta x)`

Distribute:

`m=((4x^2+8x\Delta x-4\Delta x^2-5x-5\Delta x-1)-(4x^2-5x-1))/(\Delta x)`

Distribute the right:

`m=((4x^2+8x\Delta x-4\Delta x^2-5x-5\Delta x-1)+(-4x^2+5x+1))/(\Delta x)`

Combine like terms:

`m=((4x^2-4x^2)+(-5x+5x)+(1-1)+(8x\Delta x-4\Delta x^2-5\Delta x))/(\Delta x)`

The first three terms will cancel. This leaves us with:

`m=(8x\Delta x-4\Delta x^2-5\Delta x)/(\Delta x)`

We can factor out a Δx from the numerator:

`m=(\Delta x(8x-4\Delta x-5))/(\Delta x)`

The Δx will cancel. So, our slope function is:

`m=8x-5-4\Delta x`

And we're done!