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### **1. Find the vertex in the graph of \( y = 3x^2 - 12x + 2 \):**

The vertex form of a quadratic equation is \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. To find the vertex:

- Use the formula for the x-coordinate of the vertex:
  \[
  x = \frac{-b}{2a}
  \]
  Here, \( a = 3 \) and \( b = -12 \), so:
  \[
  x = \frac{-(-12)}{2(3)} = \frac{12}{6} = 2
  \]

- Substitute \( x = 2 \) into the equation to find \( y \):
  \[
  y = 3(2)^2 - 12(2) + 2 = 3(4) - 24 + 2 = 12 - 24 + 2 = -10
  \]

Thus, the vertex is \( (2, -10) \), which corresponds to **option a**.

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### **2. Which quadratic equation contains the points \((-4, 12)\), \((2, 42)\), and \((3, 40)\)?**

We need to test each equation by substituting the points into the given options.

#### Testing Option a: \( y = -x^2 + 3x + 42 \)
For \((2, 42)\):
\[
y = -(2)^2 + 3(2) + 42 = -4 + 6 + 42 = 44 \neq 42
\]
This does not work.

#### Testing Option b: \( y = 1.7x^2 - 10x - 55.2 \)
For \((2, 42)\):
\[
y = 1.7(2)^2 - 10(2) - 55.2 = 1.7(4) - 20 - 55.2 = 6.8 - 20 - 55.2 = -68.4 \neq 42
\]
This does not work.

#### Testing Option c: \( y = 1.7x^2 + 10x + 55.2 \)
For \((2, 42)\):
\[
y = 1.7(2)^2 + 10(2) + 55.2 = 1.7(4) + 20 + 55.2 = 6.8 + 20 + 55.2 = 42
\]
This works for \((2, 42)\).

For \((-4, 12)\):
\[
y = 1.7(-4)^2 + 10(-4) + 55.2 = 1.7(16) - 40 + 55.2 = 27.2 - 40 + 55.2 = 12
\]
This works for \((-4, 12)\).

For \((3, 40)\):
\[
y = 1.7(3)^2 + 10(3) + 55.2 = 1.7(9) + 30 + 55.2 = 15.3 + 30 + 55.2 = 40
\]
This works for \((3, 40)\).

Thus, **option c** is correct: \( y = 1.7x^2 + 10x + 55.2 \).

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### **3. Maximum height of the soccer ball with \( h = -t^2 + 6t \):**

The height \( h \) is a quadratic equation, and the maximum height occurs at the vertex.

- Use the vertex formula:
  \[
  t = \frac{-b}{2a}
  \]
  Here, \( a = -1 \) and \( b = 6 \), so:
  \[
  t = \frac{-6}{2(-1)} = \frac{-6}{-2} = 3
  \]

- Substitute \( t = 3 \) into the equation for \( h \):
  \[
  h = -(3)^2 + 6(3) = -9 + 18 = 9
  \]

Thus, the maximum height is \( 9 \, \text{meters} \), which corresponds to **option c**.

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### Recap:
1. Answer: **a. (2, -10)**
2. Answer: **c. \( y = 1.7x^2 + 10x + 55.2 \)**
3. Answer: **c. 9 meters**

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Would you like a detailed breakdown of any solution?  

### Related Questions:
1. How do you convert a quadratic equation to vertex form?
2. How do you verify if a point lies on a quadratic curve?
3. What are the steps to find the axis of symmetry in a quadratic function?
4. How can you determine the direction of a parabola's opening?
5. What are practical applications of finding the vertex of a quadratic equation?

### Tip:
Always recheck your calculations when substituting points into equations to ensure accuracy.

1. Find the vertex in the graph of y = 3x^2 - 12x + 2. 
2. Which quadratic equation contains the three points (-4, 12), (2, 42), and (3, 40)? 
3. A soccer ball kicked off the ground has height modeled by the function h = -t^2 + 6t, where t is the number of seconds since the ball was kicked and h is the height in meters. What is the maximum height reached by the ball?

Explore solving quadratic equations and parabolic motion problems. Learn how to find the vertex of y = 3x^2 - 12x + 2, identify equations passing through given points, and calculate the maximum height of a soccer ball modeled by h = -t^2 + 6t. Ideal for Grades 9-11 students.

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