Exploring Vertex: Finding the Maximum of y = -2x(x+1)

Q: How do I remember which way the parabola opens?

Think of it like a smile or frown! When the coefficient of x² is positive, the parabola is like a smile (opens up, has a minimum). When negative, it's like a frown (opens down, has a maximum).

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

Not always, but it helps! In $ y = -2x(x+1) $, you can see the negative sign in front, so you know it opens downward. Expanding to $ y = -2x^2 - 2x $ confirms a = -2

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

A minimum point is the lowest point on the graph (bottom of a U-shape). A maximum point is the highest point (top of an upside-down U). The vertex is always one or the other!

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

No! Every parabola has exactly one vertex that is either a minimum OR a maximum, never both. The sign of the leading coefficient determines which one it is.

Q: How do I find the actual coordinates of the maximum point?

Use the vertex formula! For $ y = ax^2 + bx + c $, the x-coordinate is $ x = -\frac{b}{2a} $. Then substitute this x-value back into the equation to find the y-coordinate.

Quadratic Functions with Negative Leading Coefficients

Does the parable

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

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?

00:04 Let's properly open the brackets and multiply by each factor

00:18 The coefficient A of the function is negative, therefore the parabola is sad

00:23 Therefore the parabola has a maximum point

00:26 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

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

Is there a minimum or maximum point?

Step-by-step solution

To determine whether there is a minimum or maximum point in the quadratic function $y = -2x(x+1)$ , let's follow these steps:

Simplify the given function. Distribute the negative sign to expand the polynomial:

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

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

Identify the coefficients of the quadratic equation in the form $ax^2 + bx + c$ :
$a = -2$ , $b = -2$ , and $c = 0$ .
Since the coefficient of $x^2$ , $a$ , is negative, the parabola opens downwards.

When a parabola opens downwards, it has a maximum point, as it reaches a peak value (the highest point) before descending.

Therefore, the quadratic function $y = -2x(x+1)$ has a maximum point.

Hence, the correct answer is that the parable has a highest point.

Final Answer

Highest point

Key Points to Remember

Essential concepts to master this topic

Sign Rule: Negative coefficient of x² means parabola opens downward
Technique: Expand -2x(x+1) = -2x² - 2x to identify a = -2
Check: When a < 0, parabola has maximum point at vertex ✓

Common Mistakes

Avoid these frequent errors

Confusing minimum and maximum based on coefficient sign
Don't assume positive coefficient always means maximum! When a > 0, parabola opens upward (has minimum), when a < 0, opens downward (has maximum). Always check the sign of the x² coefficient to determine if the vertex is a minimum or maximum point.

Practice Quiz

Test your knowledge with interactive questions

What is the value of the coefficient $$ c $$ in the equation below?

$$ 4x^2+9x-2 $$

FAQ

Everything you need to know about this question

How do I remember which way the parabola opens?

Think of it like a smile or frown! When the coefficient of x² is positive, the parabola is like a smile (opens up, has a minimum). When negative, it's like a frown (opens down, has a maximum).

Do I always need to expand the factored form?

Not always, but it helps! In $y = -2x(x+1)$ , you can see the negative sign in front, so you know it opens downward. Expanding to $y = -2x^2 - 2x$ confirms a = -2 < 0.

What's the difference between minimum and maximum points?

A minimum point is the lowest point on the graph (bottom of a U-shape). A maximum point is the highest point (top of an upside-down U). The vertex is always one or the other!

Can a parabola have both a minimum and maximum?

No! Every parabola has exactly one vertex that is either a minimum OR a maximum, never both. The sign of the leading coefficient determines which one it is.

How do I find the actual coordinates of the maximum point?

Use the vertex formula! For $y = ax^2 + bx + c$ , the x-coordinate is $x = -\frac{b}{2a}$ . Then substitute this x-value back into the equation to find the y-coordinate.

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