Question

name:
period:
algebra
spiral 19
directions: answer each of the following questions. show all work.

1. which is an equation of the parabola graphed in the accompanying diagram?

a. $y = x^2 + 4$
b. $y = x^2 - 4$
c. $y = -x^2 + 4$
d. $y = -x^2 - 4$

1. the roots of a quadratic equation can be found using the graph below. what are the roots of this equation?

a. -4, only
b. -4 and -1
c. -1 and 4
d. -4, -1, and 4

1. what are the coordinates of the point where the graph of the equation $x + 2y = 8$ crosses the y - axis?

a. (0, 8)
b. (8, 0)
c. (0, 4)
d. (4, 0)

1. the expression $sqrt{27} + sqrt{12}$ is equal to

a. $13sqrt{3}$
b. $5sqrt{3}$
c. $5sqrt{6}$
d. $sqrt{39}$

1. a family is traveling from their home to a vacation resort hotel. the table below shows their distance from home as a function of time.

time (hrs)	0	2	5	7

determine the average rate of change between hour 2 and hour 7, including units.

1. solve using the quadratic formula: $y = 2x^2 - 2x - 3$

Explanation:

Question 1

Step1: Identify parabola direction/vertex

The parabola opens downward (so coefficient of $x^2$ is negative) and has vertex at $(0, 4)$. The vertex form is $y=ax^2+k$, where $k=4$, $a=-1$.

Step2: Match to equation

Substitute values: $y=-x^2+4$

Question 2

Step1: Define roots of quadratic

Roots are x-intercepts (where graph crosses x-axis).

Step2: Identify intercepts from graph

The graph crosses x-axis at $x=-1$ and $x=4$.

Question 3

Step1: Set x=0 for y-axis cross

To find y-intercept, substitute $x=0$ into $x+2y=8$.
<Expression>
$0 + 2y = 8$
</Expression>

Step2: Solve for y

Divide both sides by 2:
<Expression>
$y = \frac{8}{2} = 4$
</Expression>
Coordinates are $(0,4)$.

Question 4

Step1: Simplify each radical

Break down radicals into perfect squares:
<Expression>
$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$, $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$
</Expression>

Step2: Add simplified radicals

Combine like terms:
<Expression>
$3\sqrt{3} + 2\sqrt{3} = 5\sqrt{3}$
</Expression>

Question 5

Step1: Recall average rate formula

Average rate of change = $\frac{\text{Change in distance}}{\text{Change in time}}$

Step2: Substitute values from table

Time changes from 2 to 7 hours, distance from 140 to 480 mi:
<Expression>
$\frac{480 - 140}{7 - 2} = \frac{340}{5} = 68$
</Expression>
Units are miles per hour.

Question 6

Step1: Identify quadratic formula

For $ax^2+bx+c=0$, roots are $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. For $y=2x^2-2x-3$, $a=2$, $b=-2$, $c=-3$.

Step2: Calculate discriminant

<Expression>
$b^2-4ac = (-2)^2 - 4(2)(-3) = 4 + 24 = 28$
</Expression>

Step3: Substitute into formula

<Expression>
$x = \frac{-(-2)\pm\sqrt{28}}{2(2)} = \frac{2\pm2\sqrt{7}}{4}$
</Expression>

Step4: Simplify the expression

Factor out 2 in numerator:
<Expression>
$x = \frac{2(1\pm\sqrt{7})}{4} = \frac{1\pm\sqrt{7}}{2}$
</Expression>

Answer:

1. c. $y = -x^2 + 4$
2. c. -1 and 4
3. c. (0, 4)
4. b. $5\sqrt{3}$
5. 68 miles per hour
6. $x=\frac{1+\sqrt{7}}{2}$ and $x=\frac{1-\sqrt{7}}{2}$