Question

Helpppp fastttt Plz show allll workkkkk​

Answer 1

Hopefully that helps

Answer 2

Answer:

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

Step-by-step explanation:

We have the function:

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

And we want to use the difference quotient:

$m=\frac{f(x+\Delta x)-f(x)}{\Delta x}$

To find the slope function.

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

$m=\frac{(4(x+\Delta x)^2-5(x+\Delta x)-1)-(4x^2-5x-1))}{\Delta x}$

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

$m=\frac{(4(x^2+2x\Delta x-\Delta x^2)-5(x+\Delta x)-1)-(4x^2-5x-1))}{\Delta x}$

Distribute:

$m=\frac{(4x^2+8x\Delta x-4\Delta x^2-5x-5\Delta x-1)-(4x^2-5x-1)}{\Delta x}$

Distribute the right:

$m=\frac{(4x^2+8x\Delta x-4\Delta x^2-5x-5\Delta x-1)+(-4x^2+5x+1)}{\Delta x}$

Combine like terms:

$m=\frac{(4x^2-4x^2)+(-5x+5x)+(1-1)+(8x\Delta x-4\Delta x^2-5\Delta x)}{\Delta x}$

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

$m=\frac{8x\Delta x-4\Delta x^2-5\Delta x}{\Delta x}$

We can factor out a Δx from the numerator:

$m=\frac{\Delta x(8x-4\Delta x-5)}{\Delta x}$

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

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

And we're done!