Uncovering the Vertex: Analyzing y = (x+1)(-x-1) for Extremes

Q: How can I tell if a parabola has a maximum or minimum without graphing?

Look at the leading coefficient (the number in front of $ x^2 $)! If it's positive, the parabola opens upward and has a minimum. If it's negative, it opens downward and has a maximum.

Q: Why do I need to expand the factored form first?

The factored form $ (x+1)(-x-1) $ hides the leading coefficient. You must expand to get standard form $ ax^2 + bx + c $ to see the sign of $ a $.

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

A maximum point is the highest point on the parabola (like a mountain peak), while a minimum point is the lowest point (like a valley). The vertex is always one or the other!

Q: Can a parabola have both a maximum and minimum?

No! Every parabola has exactly one vertex that is either the highest point OR the lowest point, never both. The shape of a parabola prevents it from having multiple extremes.

Q: What if the leading coefficient is zero?

If the coefficient of $ x^2 $ equals zero, then it's not a quadratic function anymore! You'd have a linear function instead, which doesn't form a parabola.

Q: How do I expand (x+1)(-x-1) correctly?

Use the distributive property: $ (x+1)(-x-1) = x(-x-1) + 1(-x-1) = -x^2 - x - x - 1 = -x^2 - 2x - 1 $. Watch the signs carefully!

Quadratic Functions with Factored Form 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

00:08 Does the parabola have a highest or lowest point?

00:11 Alright, first let's expand the parentheses by multiplying each factor together.

00:20 Next, let's gather all the like terms. This will simplify our expression.

00:29 Since the coefficient, A, is negative, the parabola opens downward. Think of it as a sad face.

00:35 So, the parabola has a maximum point at the top.

00:39 And that's how we solve this problem!

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, follow these steps:

Step 1: Expand the quadratic function.
Step 2: Determine the nature based on the leading coefficient.

Step 1: Expand the quadratic function:

The given function is $y = (x+1)(-x-1)$ .

Expanding this, we have:

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

Step 2: Determine the nature using the leading coefficient:

The quadratic function is $y = -x^2 - 2x - 1$ .

Here, the leading coefficient $a = -1$ . Since the leading coefficient is negative, the parabola opens downwards.

Therefore, the quadratic function has a highest point, or a maximum.

Thus, the solution to the problem is that the quadratic function has a highest point.

Final Answer

Highest point

Key Points to Remember

Essential concepts to master this topic

Leading Coefficient Rule: Negative coefficient means parabola opens downward, creating maximum
Expansion Technique: $(x+1)(-x-1) = -x^2 - 2x - 1$ gives $a = -1$
Verification: Check that $a < 0$ confirms downward opening and maximum point ✓

Common Mistakes

Avoid these frequent errors

Assuming all parabolas have minimum points
Don't automatically think every parabola has a lowest point = wrong conclusion about vertex nature! The sign of the leading coefficient determines whether the parabola opens up (minimum) or down (maximum). Always expand the function and check if the coefficient of $x^2$ is positive or negative.

Practice Quiz

Test your knowledge with interactive questions

Identify the coefficients based on the following equation

$$ y=x^2 $$

FAQ

Everything you need to know about this question

How can I tell if a parabola has a maximum or minimum without graphing?

Look at the leading coefficient (the number in front of $x^2$ )! If it's positive, the parabola opens upward and has a minimum. If it's negative, it opens downward and has a maximum.

Why do I need to expand the factored form first?

The factored form $(x+1)(-x-1)$ hides the leading coefficient. You must expand to get standard form $ax^2 + bx + c$ to see the sign of $a$ .

What's the difference between a maximum and minimum point?

A maximum point is the highest point on the parabola (like a mountain peak), while a minimum point is the lowest point (like a valley). The vertex is always one or the other!

Can a parabola have both a maximum and minimum?

No! Every parabola has exactly one vertex that is either the highest point OR the lowest point, never both. The shape of a parabola prevents it from having multiple extremes.

What if the leading coefficient is zero?

If the coefficient of $x^2$ equals zero, then it's not a quadratic function anymore! You'd have a linear function instead, which doesn't form a parabola.

How do I expand (x+1)(-x-1) correctly?

Use the distributive property: $(x+1)(-x-1) = x(-x-1) + 1(-x-1) = -x^2 - x - x - 1 = -x^2 - 2x - 1$ . Watch the signs carefully!

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Uncovering the Vertex: Analyzing y = (x+1)(-x-1) for Extremes

Quadratic Functions with Factored Form 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 can I tell if a parabola has a maximum or minimum without graphing?

Why do I need to expand the factored form first?

What's the difference between a maximum and minimum point?

Can a parabola have both a maximum and minimum?

What if the leading coefficient is zero?

How do I expand (x+1)(-x-1) correctly?

🌟 Unlock Your Math Potential

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