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

Does the parable

y=(x+1)(−x−1) 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
1

Understand the problem

Does the parable

y=(x+1)(−x−1) y=(x+1)(-x-1)

Is there a minimum or maximum point?

2

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

Expanding this, we have:

y=(x+1)(−x−1)=x(−x−1)+1(−x−1)=−x2−x−x−1=−x2−2x−1 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=−x2−2x−1 y = -x^2 - 2x - 1 .

Here, the leading coefficient a=−1 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.

3

Final Answer

Highest point

Practice Quiz

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

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

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