Answer:
[tex]m=8x-5-4\Delta x[/tex]
Step-by-step explanation:
We have the function:
[tex]f(x)=4x^2-5x-1[/tex]
And we want to use the difference quotient:
[tex]m=\frac{f(x+\Delta x)-f(x)}{\Delta x}[/tex]
To find the slope function.
So, let's substitute [tex]x+\Delta x[/tex] for our function. This yields:
[tex]m=\frac{(4(x+\Delta x)^2-5(x+\Delta x)-1)-(4x^2-5x-1))}{\Delta x}[/tex]
Let's square. Use the perfect square trinomial pattern. This yields:
[tex]m=\frac{(4(x^2+2x\Delta x-\Delta x^2)-5(x+\Delta x)-1)-(4x^2-5x-1))}{\Delta x}[/tex]
Distribute:
[tex]m=\frac{(4x^2+8x\Delta x-4\Delta x^2-5x-5\Delta x-1)-(4x^2-5x-1)}{\Delta x}[/tex]
Distribute the right:
[tex]m=\frac{(4x^2+8x\Delta x-4\Delta x^2-5x-5\Delta x-1)+(-4x^2+5x+1)}{\Delta x}[/tex]
Combine like terms:
[tex]m=\frac{(4x^2-4x^2)+(-5x+5x)+(1-1)+(8x\Delta x-4\Delta x^2-5\Delta x)}{\Delta x}[/tex]
The first three terms will cancel. This leaves us with:
[tex]m=\frac{8x\Delta x-4\Delta x^2-5\Delta x}{\Delta x}[/tex]
We can factor out a Δx from the numerator:
[tex]m=\frac{\Delta x(8x-4\Delta x-5)}{\Delta x}[/tex]
The Δx will cancel. So, our slope function is:
[tex]m=8x-5-4\Delta x[/tex]
And we're done!
