Analyze Function Behavior: Multiplication with Opposite Signs

Q: How can a function be both increasing and decreasing?

A function can have different behaviors on different intervals! For $ f(x) = -x^2 $, it increases when x 0, with a maximum at x = 0.

Q: Why does multiplying by the opposite sign create this pattern?

When you multiply x by (-x), you get $ -x^2 $. This creates a parabola opening downward, which naturally increases on the left side and decreases on the right side of the vertex.

Q: What's the difference between this and a regular parabola?

This function $ f(x) = -x^2 $ is a downward-opening parabola with vertex at (0,0). Regular parabolas like $ y = x^2 $ open upward and are decreasing then increasing.

Q: How do I find where the function changes from increasing to decreasing?

Look for the vertex or turning point! For $ f(x) = -x^2 $, this happens at x = 0 where f(0) = 0, which is the maximum point.

Q: Can I determine this without plotting points?

Yes! Since $ f(x) = -x^2 $ is a quadratic with negative coefficient, you know it's a downward parabola that increases then decreases, with maximum at x = 0.

Quadratic Functions with Sign Analysis

Determine whether the function described below is increasing, decreasing, or constant:

Each number is multiplied by the same number with different signs.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is the function increasing, decreasing, or constant?

00:03 Let's draw the function graph

00:13 Let's substitute X values and find the corresponding Y values

00:26 Now let's substitute negative X values to check the left side of the graph

00:54 We can see that the function increases until the origin and then decreases

00:58 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

Determine whether the function described below is increasing, decreasing, or constant:

Each number is multiplied by the same number with different signs.

Step-by-step solution

The function is:

$f(x)=x\times(-x)$

Let's start by assuming that $x$ equals 0:

$f(0)=0\times0=0$

Now let's assume that $x$ equals 1:

$f(1)=1\times-1=-1$

Now let's assume that $x$ equals -1:

$f(-1)=(-1)\times(-1)=1$

Now let's assume that $x$ equals 2:

$f(2)=2\times(-2)=-4$

Now let's assume that $x$ equals -2:

$f(-2)=(-2)\times(-2)=4$

Finally, let's plot all of the points on a graph:

We can see that the function both increases and decreases.

Final Answer

Increasing and decreasing

Key Points to Remember

Essential concepts to master this topic

Function Form: f(x) = x × (-x) = -x² creates parabola opening downward
Technique: Test multiple points: f(-2) = 4, f(0) = 0, f(2) = -4
Check: Graph shows increasing left of origin, decreasing right of origin ✓

Common Mistakes

Avoid these frequent errors

Assuming function has constant behavior throughout domain
Don't conclude f(x) = -x² is only decreasing because some values decrease! The function increases for x < 0 and decreases for x > 0. Always test points on both sides of key values like x = 0.

Practice Quiz

Test your knowledge with interactive questions

Does the function in the graph decrease throughout?

FAQ

Everything you need to know about this question

How can a function be both increasing and decreasing?

A function can have different behaviors on different intervals! For $f(x) = -x^2$ , it increases when x < 0 and decreases when x > 0, with a maximum at x = 0.

Why does multiplying by the opposite sign create this pattern?

When you multiply x by (-x), you get $-x^2$ . This creates a parabola opening downward, which naturally increases on the left side and decreases on the right side of the vertex.

What's the difference between this and a regular parabola?

This function $f(x) = -x^2$ is a downward-opening parabola with vertex at (0,0). Regular parabolas like $y = x^2$ open upward and are decreasing then increasing.

How do I find where the function changes from increasing to decreasing?

Look for the vertex or turning point! For $f(x) = -x^2$ , this happens at x = 0 where f(0) = 0, which is the maximum point.

Can I determine this without plotting points?

Yes! Since $f(x) = -x^2$ is a quadratic with negative coefficient, you know it's a downward parabola that increases then decreases, with maximum at x = 0.

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