Question

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

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:

$\frac{f(b) - f(a)}{b - a}$

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

$= \frac{f(6) - f(2)}{6 - 2} \\ = \frac{ {6}^{2} + 3 \times 6 - 2 - {2}^{2} - 3 \times 2 + 2 }{4} \\ = \frac{36 + 18 - 4 - 6}{4} \\ = \frac{44}{4} \\ = 11$

The average rate of change is 11