Analyzing Maximum or Minimum in the Parabola: y = -(x+1)(x-1)

Q: How do I know if a parabola opens up or down?

Look at the coefficient of $ x^2 $! If it's positive, the parabola opens upward (U-shape). If it's negative, it opens downward (∩-shape).

Q: What's the difference between maximum and minimum points?

A maximum point is the highest point on the graph (peak), while a minimum point is the lowest point (valley). Think of it like a mountain top versus a valley bottom!

Q: Do I always need to expand the factored form?

Not always, but it helps! In factored form like $ -(x+1)(x-1) $, you can see the negative sign in front tells you a is negative, so it opens downward.

Q: Can a parabola have both maximum AND minimum points?

No! Every parabola has exactly one vertex that is either a maximum OR a minimum point, never both. The direction of opening determines which one it is.

Q: What if the coefficient of x² is zero?

If the coefficient of $ x^2 $ is zero, then it's not a quadratic function anymore! It becomes a linear function (straight line) with no maximum or minimum point.

Quadratic Functions with Vertex Analysis

Does the parable

$y=-(x+1)(x-1)$

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

00:00 Does the parabola have a maximum or minimum point?

00:03 We'll use the shortened multiplication formulas and expand the brackets

00:12 Negative times positive is always negative

00:15 Negative times negative is always positive

00:21 The coefficient A of the function is negative, therefore the parabola is sad

00:24 Therefore the parabola has a maximum point

00:28 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Does the parable

$y=-(x+1)(x-1)$

Is there a minimum or maximum point?

Step-by-step solution

To solve this problem, let's analyze the function step-by-step:

Step 1: Expand the given quadratic function.
The function provided is $y = -(x+1)(x-1)$ . We rewrite it by expanding:

$y = -(x^2 - 1) = -x^2 + 1$ .

Step 2: Determine the direction of the parabola.
The standard form $y = ax^2 + bx + c$ indicates that the parabola opens upwards if $a > 0$ and downwards if $a < 0$ . Here, the value of $a$ is $-1$ , which means the parabola opens downwards.

Step 3: Identify the vertex type.
Since the parabola opens downwards, the vertex represents the highest point on the graph, which is a maximum.

Therefore, the parabola has a maximum point. The correct choice is:

Choice 2: Highest point

Thus, we conclude that the given quadratic function has a maximum point.

Final Answer

Highest point

Key Points to Remember

Essential concepts to master this topic

Rule: Coefficient of $x^2$ determines parabola direction
Technique: Expand $-(x+1)(x-1) = -x^2 + 1$ to identify coefficient
Check: Since $a = -1 < 0$ , parabola opens downward = maximum point ✓

Common Mistakes

Avoid these frequent errors

Confusing upward/downward direction with min/max
Don't think upward parabola has maximum point = wrong conclusion! Upward opening means the vertex is the lowest point (minimum). Always remember: downward opening (a < 0) gives maximum, upward opening (a > 0) gives 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 know if a parabola opens up or down?

Look at the coefficient of $x^2$ ! If it's positive, the parabola opens upward (U-shape). If it's negative, it opens downward (∩-shape).

What's the difference between maximum and minimum points?

A maximum point is the highest point on the graph (peak), while a minimum point is the lowest point (valley). Think of it like a mountain top versus a valley bottom!

Do I always need to expand the factored form?

Not always, but it helps! In factored form like $-(x+1)(x-1)$ , you can see the negative sign in front tells you a is negative, so it opens downward.

Can a parabola have both maximum AND minimum points?

No! Every parabola has exactly one vertex that is either a maximum OR a minimum point, never both. The direction of opening determines which one it is.

What if the coefficient of x² is zero?

If the coefficient of $x^2$ is zero, then it's not a quadratic function anymore! It becomes a linear function (straight line) with no maximum or minimum point.

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Analyzing Maximum or Minimum in the Parabola: y = -(x+1)(x-1)

Quadratic Functions with Vertex Analysis

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know if a parabola opens up or down?

What's the difference between maximum and minimum points?

Do I always need to expand the factored form?

Can a parabola have both maximum AND minimum points?

What if the coefficient of x² is zero?

🌟 Unlock Your Math Potential

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