Function Behavior Analysis: Effects of Constant Multiplication

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

A function like $ f(x) = x^2 $ changes behavior at its vertex! It decreases for x 0. This makes it both increasing and decreasing overall.

Q: Where exactly does the function stop decreasing and start increasing?

At the vertex! For $ f(x) = x^2 $, this happens at x = 0. The function decreases until x = 0, then increases after x = 0.

Q: Do I need to test negative numbers too?

Yes! You must test values on both sides of the vertex to see the complete behavior. Testing only positive numbers would miss half the story!

Q: What if I plotted the points wrong?

Double-check your calculations: $ (-2)^2 = 4 $, $ (-1)^2 = 1 $, $ 0^2 = 0 $. Remember that negative numbers squared become positive!

Q: Is this the same for all parabolas?

All parabolas that open upward (like $ y = x^2 $) are both decreasing and increasing. Parabolas that open downward are both increasing and decreasing in the opposite pattern.

Quadratic Functions with Monotonicity Analysis

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

Each number is multiplied by the same number.

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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

00:40 Now let's substitute negative X values to check the left side of the graph

01:02 We can see that the graph decreases until the origin and then increases

01:10 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.

Step-by-step solution

The function is:

$f(x)=x^2$

Let's start with x equal to 0:

$f(0)=0^2=0$

Now let's assume x is equal to 1:

$f(1)=1^2=1$

Now let's assume x is equal to 2:

$f(2)=2^2=4$

Now let's assume x is equal to -1:

$f(-1)=(-1)^2=1$

Now let's assume x is equal to -2:

$f(-2)=(-2)^2=4$

Let's finally plot all the points on the graph:

We can see that we have a function that is both increasing and decreasing.

Final Answer

Increasing and decreasing

Key Points to Remember

Essential concepts to master this topic

Parabola Behavior: Quadratic functions have vertex as turning point
Test Method: Check values: f(-2) = 4, f(-1) = 1, f(0) = 0, f(1) = 1, f(2) = 4
Verification: Plot points and observe U-shape: decreasing left, increasing right ✓

Common Mistakes

Avoid these frequent errors

Assuming quadratic functions have constant behavior
Don't say a quadratic is only increasing or only decreasing = wrong classification! Quadratics change from decreasing to increasing (or vice versa) at their vertex. Always check both sides of the vertex to see the complete behavior.

Practice Quiz

Test your knowledge with interactive questions

Is the function in the graph decreasing?

FAQ

Everything you need to know about this question

How can a function be both increasing AND decreasing?

A function like $f(x) = x^2$ changes behavior at its vertex! It decreases for x < 0 and increases for x > 0. This makes it both increasing and decreasing overall.

Where exactly does the function stop decreasing and start increasing?

At the vertex! For $f(x) = x^2$ , this happens at x = 0. The function decreases until x = 0, then increases after x = 0.

Do I need to test negative numbers too?

Yes! You must test values on both sides of the vertex to see the complete behavior. Testing only positive numbers would miss half the story!

What if I plotted the points wrong?

Double-check your calculations: $(-2)^2 = 4$ , $(-1)^2 = 1$ , $0^2 = 0$ . Remember that negative numbers squared become positive!

Is this the same for all parabolas?

All parabolas that open upward (like $y = x^2$ ) are both decreasing and increasing. Parabolas that open downward are both increasing and decreasing in the opposite pattern.

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