Question

1. What are the solutions (coordinate points) to the system of equations?

y=x^2+5x+6 and y=3x+6

2. Prove algebraically what type of function this is (even, odd, or neither).
f(x)=x^6-x^4

3. Given the function f(x)=x^2+3x-2
What is the average rate of change for the function from 2 to 6? Show your work.

Answer 1

Question 1

The given system of equations is:

`y = {x}^(2) + 5x + 6 \\ y = 3x + 6`

Equate the two equations:

`{x}^(2) + 5x + 6 = 3x + 6`

Rewrite in standard form:

`{x}^(2) + 5x - 3x + 6 - 6 = 0`

`{x}^(2) + 2x = 0`

`x(x + 2) = 0`

`x = 0 \: or \: x = - 2`

When we put x=0, in y=3x +6, we get:

`y = 3(0) + 6 = 6`

One solution is (0,6)

When we put x=-2, into y=3x+6, we get:

`y = 3( - 2) + 6 = 0`

Another solution is (-2,0)

The solutions are; (0,6) and (-2,0)

Question 2:

The function is

`f(x) = {x}^(6) - {x}^(4)`

Let us put x=-x,

`f( - x) = {( - x)}^(6) - {( - x)}^(4)`

This gives:

`f( - x) = {x}^(6) - {x}^(4)`

We can observe that:

`f(x) = f( - x)`

This is the property of an even function.

Question 3:

The given function is

`f(x) = {x}^(2) + 3x - 2`

The average rate of change of f(x) from x=a to x=b is given as:

`(f(b) - f(a))/(b - a)`

This is the slope of the secant line connecting the two points on f(x)

From x=2 to x=6, the average rate of change

`= (f(6) - f(2))/(6 - 2) \\ = \frac{ {6}^(2) + 3 * 6 - 2 - {2}^(2) - 3 * 2 + 2 }{4} \\ = (36 + 18 - 4 - 6)/(4) \\ = (44)/(4) \\ = 11`

The average rate of change is 11