Explore the Quadratic Vertex: Does y = -x² + 3x + 9 Have a Maximum or Minimum?

Quadratic Functions with Coefficient Sign Analysis

Does the parable

y=x2+3x+9 y=-x^2+3x+9

Is there a minimum or maximum point?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:07 Does the parabola have a maximum or minimum point?
00:15 The coefficient, A, in front of the X squared term is negative. So, the parabola is upside down or sad.
00:25 This means the parabola has a maximum point.
00:29 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Does the parable

y=x2+3x+9 y=-x^2+3x+9

Is there a minimum or maximum point?

2

Step-by-step solution

The quadratic function is given by y=x2+3x+9 y = -x^2 + 3x + 9 .

In the general quadratic form y=ax2+bx+c y = ax^2 + bx + c , the coefficient a a determines the parabola's orientation:

  • If a>0 a > 0 : The parabola opens upwards, indicating a minimum point.
  • If a<0 a < 0 : The parabola opens downwards, indicating a maximum point.

For this function, a=1 a = -1 . Since a<0 a < 0 , the parabola opens downwards.

Therefore, the function has a maximum point.

Thus, the correct answer is the function has a highest point.

Therefore, the solution to the problem is choice 2: Highest point.

3

Final Answer

Highest point

Key Points to Remember

Essential concepts to master this topic
  • Rule: Coefficient a a determines parabola direction: negative opens downward
  • Technique: For y=x2+3x+9 y = -x^2 + 3x + 9 , identify a=1<0 a = -1 < 0
  • Check: Negative a a means downward opening = maximum point ✓

Common Mistakes

Avoid these frequent errors
  • Confusing coefficient sign with vertex location
    Don't think positive coefficient means maximum point = wrong conclusion! The sign of a a only tells you parabola direction, not vertex height. Always remember: negative a a opens downward (maximum), positive a a opens upward (minimum).

Practice Quiz

Test your knowledge with interactive questions

What is the value of the coefficient \( b \) in the equation below?

\( 3x^2+8x-5 \)

FAQ

Everything you need to know about this question

How do I remember which direction the parabola opens?

+

Think of it like a smile or frown! When a>0 a > 0 , the parabola is a smile (opens up) with a minimum. When a<0 a < 0 , it's a frown (opens down) with a maximum.

What if the coefficient a is really small, like -0.1?

+

Size doesn't matter - only the sign! Whether a=1 a = -1 or a=0.1 a = -0.1 , both are negative, so both parabolas open downward and have maximum points.

Do I need to find the actual vertex to answer this question?

+

No! You only need the coefficient a a . For y=x2+3x+9 y = -x^2 + 3x + 9 , since a=1<0 a = -1 < 0 , you know it has a maximum without calculating where.

What's the difference between maximum and minimum points?

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A maximum point is the highest point on the parabola (like the top of a hill). A minimum point is the lowest point (like the bottom of a valley). The vertex is always one or the other!

Can a parabola have both a maximum AND minimum?

+

Never! Each parabola has exactly one vertex that is either a maximum point or a minimum point, but not both. The sign of a a determines which one.

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