Step-by-step explanation:
6 and 7 was already answered so I'll do 5 and 8.
5. The point the curve crosses the x axis is the x intercept, to find the x intercept of a rational function, we set y=0, and solve for x.
[tex]0 = \frac{x - 4}{x} [/tex]
Set the numerator equal to 0.
[tex]x - 4 = 0[/tex]
[tex]x = 4[/tex]
So the x intercept is (4,0).
Next, to find gradient, we take the derivative of the function.
[tex] \frac{x - 4}{x} [/tex]
We could use product rule, but for simplicity, serperate the function
[tex] \frac{x}{x} - \frac{4}{x} [/tex]
[tex]1 - \frac{4}{x} [/tex]
Next using exponents rules,
[tex]1 - 4 {x}^{ - 1} [/tex]
Now we take the derivative,
Derivative of a constant is zero.
Derivative of a power function,
[tex]n \times x {}^{n - 1} [/tex]
We move the exponent to the front, then we subtract the exponent by 1.
So, we get
[tex]4 {x}^{ - 2} [/tex]
Now, we plug in. 4,
[tex]4(4) {}^{ - 2} [/tex]
[tex]4 \times \frac{1}{16} = \frac{1}{4} [/tex]
The slope or gradient at the x intercept is 1/4
8. The derivative of ax^2+bx, with respect to x is
[tex]2ax + b[/tex]
When x=2, we have a gradient of 8.
[tex]2a(2) + b = 8[/tex]
[tex]4a + b = 8[/tex]
When x=-1, we have a gradient of -10.
[tex]2a( - 1) + b = - 10[/tex]
[tex] - 2a + b = - 10[/tex]
We have two system of equations,
[tex]4a + b = 8[/tex]
[tex] - 2a + b = - 10[/tex]
Let subtract the system to eliminate b.
[tex]6a = 18[/tex]
[tex]a = 3[/tex]
Plug 3 for a, back in to solve for b.
[tex]4(3) + b = 8[/tex]
[tex]12 + b = 8[/tex]
[tex]b = - 4[/tex]
So a is 3
b is -4
