Lesson 2 Root Variation Solidify Understanding

Ready

A team of astronomers have been studying the paths of specific objects in the night sky by graphing them in a coordinate plane. The origin represents midnight. The $x$ value before $(0, 0)$ represents the hours before midnight and the numbers to the right of $(0, 0)$ represent the time of day. They realize that their graphs look like the systems of equations they learned about in their high school math classes. Use each graph to find the point(s) of intersection. Then state the time of day or night that the paths of the objects intersect. (Hint: An hour before midnight would show as $- 1$ in the graph and would represent 11 p.m.)

1.

$[\begin{matrix} x + 2 y = 6 \\ x - y = 3 \end{matrix}$

2.

$[\begin{matrix} 3 x + y = - 2 \\ y = x^{2} \end{matrix}$

Use Dr. Stella’s method for problems 3 and 4.

One astronomer, Dr. Stella, wrote a system of two equations to see if she could predict when the paths of the objects she was studying would intersect.

She solved the system by setting the equations equal to each other and solving for $x$ . She then used substitution to find $y$ .

After solving the systems, she looked at the graphs of the equations to check if she was right.

3.

Set the equations equal to each other and solve for $x$ .

$[\begin{matrix} y = 2 x - 1 \\ y = - x + 5 \end{matrix}$

Check your answer by graphing.

4.

Set the equations equal to each other and solve for $x$ .

$[\begin{matrix} y = x^{2} + 1 \\ y = x + 3 \end{matrix}$

Check your answers by graphing.

Set

The square root function $y = 8 \sqrt{x}$ gives the speed in feet per second of an object in free fall after falling $x$ feet on the planet Vogabah.

5.

Find the speed of an object in free fall after it has fallen $9 feet$ .

6.

Find the speed of an object in free fall after it has fallen $5 feet$ . (Round to the nearest $\frac{ft}{s} .$ )

7.

The table shows the speed of the object during the first $5 seconds$ of free fall. The object doubled its speed from $1 second$ to $4 seconds$ .

$x (ft)$	$f (\frac{ft}{sec})$
$0$	$0$
$1$	$8$
$2$	$11.3$
$3$	$13.9$
$4$	$16$
$5$	$17.9$

a.

At how many seconds will the object’s speed double again?

b.

At how many seconds will the object’s speed double a third time?

8.

If an object in free fall is moving at a speed of $48 \frac{ft}{s},$ how many feet to the nearest foot has it fallen?

9.

If an object in free fall is moving at a speed of $72 \frac{ft}{s},$ how many feet to the nearest foot has it fallen?

10.

Graph $y = \sqrt{x}$ and $y = 8 \sqrt{x}$ on the same set of axes. How are they the same, and how are they different?

11.

Two graphs are shown on the same grid. Both graphs show a proportional relationship. In one, the output quantity varies directly with the input quantity. In the other, the output quantity varies directly with the square root of the input quantity. Write the equations for each graph.

Go

Find each product.

12.

${(2 x + 5)}^{2}$

13.

${(3 x - 1)}^{2}$

14.

${(x + 7 y)}^{2}$

15.

${(a b + c)}^{2}$

16.

${(6 a^{2} b^{3} + 1)}^{2}$

17.

${(2 a^{2} b^{3} - 5)}^{2}$

18.

What is the square root of $x^{2}$ ?

Find the square root.

19.

$\sqrt{16 a^{2}}$

20.

$\sqrt{81 a^{4} b^{2}}$

21.

$\sqrt{a^{2} b^{6} c^{4} d^{10}}$