Analyze the Parabola y=(x-2)(x+1): Finding the Turning Point

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

Look at the coefficient of $ x^2 $ after expanding. If it's positive, the parabola opens upward (has a minimum). If it's negative, it opens downward (has a maximum).

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

For determining direction, yes! Expanding $ (x-2)(x+1) $ to $ x^2 - x - 2 $ shows the leading coefficient clearly. The factored form hides this crucial information.

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

A minimum point is the lowest point on the graph (like the bottom of a U-shape). A maximum point is the highest point (like the top of an upside-down U).

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

No! Every parabola has exactly one turning point called the vertex. It's either a minimum OR a maximum, never both. The direction the parabola opens determines which type it is.

Q: How do I expand (x-2)(x+1) correctly?

Use FOIL: First terms ($ x \cdot x = x^2 $), Outer terms ($ x \cdot 1 = x $), Inner terms ($ -2 \cdot x = -2x $), Last terms ($ -2 \cdot 1 = -2 $). Then combine: $ x^2 + x - 2x - 2 = x^2 - x - 2 $.

Quadratic Functions with Factored Form Analysis

Does the parable

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

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 Let's properly open parentheses, multiply each factor by each factor

00:13 Let's group the factors

00:21 The coefficient A of the function is positive, therefore the parabola is smiling

00:26 Therefore the parabola has a minimum point

00:31 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

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

Is there a minimum or maximum point?

Step-by-step solution

To determine if the function $y = (x-2)(x+1)$ has a minimum or maximum point, we start by converting it from product form to standard form:

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

Expanding the expression:

$y = x^2 + x - 2x - 2$

Simplify:

$y = x^2 - x - 2$

In standard form, $y = x^2 - x - 2$ , the coefficient of $x^2$ , which is $a = 1$ , is positive. A positive $a$ indicates the parabola opens upwards.

Since the parabola opens upwards, it has a minimal point (vertex) as its lowest point.

Therefore, the parabola $y = (x-2)(x+1)$ has a minimal point.

Final Answer

Minimal point

Key Points to Remember

Essential concepts to master this topic

Direction Rule: Positive coefficient of $x^2$ means parabola opens upward
Expansion Technique: Convert $(x-2)(x+1)$ to $x^2 - x - 2$
Verification: Check that $a = 1 > 0$ confirms minimum point exists ✓

Common Mistakes

Avoid these frequent errors

Confusing upward-opening with maximum point
Don't assume upward-opening parabolas have maximum points = completely wrong answer! When $a > 0$ , the parabola opens upward like a smile, creating a minimum (lowest) point at the vertex. Always remember: upward opening = minimum point, downward opening = maximum point.

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 do I know if a parabola opens up or down?

Look at the coefficient of $x^2$ after expanding. If it's positive, the parabola opens upward (has a minimum). If it's negative, it opens downward (has a maximum).

Do I always need to expand the factored form?

For determining direction, yes! Expanding $(x-2)(x+1)$ to $x^2 - x - 2$ shows the leading coefficient clearly. The factored form hides this crucial information.

What's the difference between minimum and maximum points?

A minimum point is the lowest point on the graph (like the bottom of a U-shape). A maximum point is the highest point (like the top of an upside-down U).

Can a parabola have both minimum and maximum points?

No! Every parabola has exactly one turning point called the vertex. It's either a minimum OR a maximum, never both. The direction the parabola opens determines which type it is.

How do I expand (x-2)(x+1) correctly?

Use FOIL: First terms ( $x \cdot x = x^2$ ), Outer terms ( $x \cdot 1 = x$ ), Inner terms ( $-2 \cdot x = -2x$ ), Last terms ( $-2 \cdot 1 = -2$ ). Then combine: $x^2 + x - 2x - 2 = x^2 - x - 2$ .

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Analyze the Parabola y=(x-2)(x+1): Finding the Turning Point

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 do I know if a parabola opens up or down?

Do I always need to expand the factored form?

What's the difference between minimum and maximum points?

Can a parabola have both minimum and maximum points?

How do I expand (x-2)(x+1) correctly?

🌟 Unlock Your Math Potential

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