Let’s analyze and solve the questions based on the given mathematical problems from the image.

6. Quadratic Function

The function is $g(x) = -(x - 2)^2 + 5$.

(a) What is the vertex of the graph of function $g$?
The vertex form of a quadratic function is written as $g(x) = a(x - h)^2 + k$, where the vertex is $(h, k)$.
Here:

• $h = 2$
• $k = 5$
  Thus, the vertex is (2, 5).

(b) Does the vertex represent the minimum value or the maximum value of the function? Explain.
The coefficient of $(x - h)^2$, which is $a = -1$, is negative. This indicates the parabola opens downward, so the vertex represents the maximum value of the function.

(c) If you were to shift this graph 3 units left from where it is now, what would be the equation represented by the new graph?
To shift the graph 3 units left, replace $x$ with $x + 3$ in the equation:
$g(x) = -(x - 2 + 3)^2 + 5$
Simplify:
$g(x) = -(x + 1)^2 + 5$

The new equation is $g(x) = -(x + 1)^2 + 5$.

7. Helicopter Height Function

The function is $h(t) = (50 + 4.9t)(8 - t)$.

(a) Find or approximate the time when the object hits the ground. Explain or show your reasoning.
The object hits the ground when the height $h(t) = 0$. Set the equation to 0:
$(50 + 4.9t)(8 - t) = 0$
This product is zero if either factor is zero:

1. $50 + 4.9t = 0$: Solve for $t$:
   $t = -\frac{50}{4.9} \approx -10.2$ (not valid because $t \geq 0$)

2. $8 - t = 0$: Solve for $t$:
   $t = 8$

Thus, the object hits the ground at $t = 8$ seconds.

(b) From what height is the object dropped?
The object is dropped at $t = 0$. Substitute $t = 0$ into $h(t)$:
$h(0) = (50 + 4.9(0))(8 - 0) = 50 \cdot 8 = 400$

The object is dropped from a height of 400 meters.

(c) When will the object reach its maximum height? Explain.
The function $h(t) = (50 + 4.9t)(8 - t)$ represents a parabola. The maximum height occurs at the vertex. Rewrite $h(t)$:
$h(t) = -4.9t^2 + (8 \cdot 4.9 - 50)t + 400$
The vertex occurs at $t = -\frac{b}{2a}$, where $a = -4.9$ and $b = (8 \cdot 4.9 - 50)$:
$b = 39.2 - 50 = -10.8$
$t = -\frac{-10.8}{2(-4.9)} = \frac{10.8}{9.8} \approx 1.1 \, \text{seconds}$

The object reaches its maximum height at approximately 1.1 seconds.

Do you want additional details or explanations?

Here are 5 related questions to extend your learning:

1. How do you determine the axis of symmetry for a parabola?
2. Why does a negative coefficient in a quadratic equation indicate a maximum value?
3. How would the equation change if the graph of $g(x)$ were shifted 5 units up?
4. Can you write the factored form of the quadratic height equation $h(t)$?
5. How would air resistance alter the function for the helicopter drop?

Tip: Always analyze the sign of the leading coefficient $a$ to understand the behavior of a parabola (opening up or down).