Increasing and Decreasing Intervals of a Parabola

Increasing and Decreasing Domain of a Parabola - Examples, Exercises and Solutions
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Increasing and Decreasing Domain of a Parabola
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Increasing and Decreasing Intervals of a Parabola

The intervals of increase and decrease describe the xx in which the parabola goes up and those in which it goes down.
Let's see it in an illustration:

B4 - The areas of increase and decrease describe the X where the parabola increases decreases

We must always observe the function from left to right.
An increasing interval is where the function's output (y-values) gets larger as we move from left to right along the x-axis, while a decreasing interval is where the y-values get smaller as x increases.
When we see a negative slope (this is how decrease looks) – the function is decreasing.
When we see a positive slope (this is how increase looks) – the function is increasing.

The parabola will change interval only at one point - at the vertex of the parabola, he highest or lowest point of the curve. Since parabolas have a characteristic U-shape (opening upward) or upside-down U-shape (opening downward), they change from decreasing to increasing (or vice versa) at exactly once at the vertex.

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Test yourself on increasing and decreasing domain of a parabola!

Note that the graph of the function below does not intersect the x-axis

Moreover the parabola's vertex is A

Identify the interval where the function is increasing:

XXXAAA

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Increasing and Decreasing Intervals of a Parabola

Understanding the Basic Concept

To examine the intervals of increase and decrease of the function, we can observe in the illustration
What happens when the x's are smaller than the XX of the vertex and what happens when the x's are larger than the XX of the vertex?
What are the intervals of increase and decrease in this example?
We can see that the XX of the vertex is −2-2.

When X>−2 X>-2    the function is increasing and, therefore, there is an interval of increase.
When X<−2  X<-2     the function is decreasing and, therefore, there is an interval of decrease.

Key Definitions

  • Increasing Interval: Where y-values get larger as x-values increase
  • Decreasing Interval: Where y-values get smaller as x-values increase
  • Vertex: The turning point where the function changes from increasing to decreasing (or vice versa)

What happens when there is no illustration?
We can examine the equation of the function and determine based on the coefficient of X2 X^2 whether it is a function with a minimum or maximum point.
When the coefficient is positive (happy face) - minimum
When the coefficient is negative (sad face) - maximum

And that's it! We have all the information to determine when it is an increasing function and when it is decreasing - the vertex of the parabola is the highest or lowest point according to the concerning parabola.
All that remains for us to do is, draw a sketch to see in it the intervals of increase and decrease clearly.

Understanding the "Happy Face" and "Sad Face"

  • Happy face (a > 0): Parabola opens upward like a smile ∪
    • Has a minimum point (vertex is the lowest point)
    • Decreases before the vertex, increases after the vertex
  • Sad face (a < 0): Parabola opens downward like a frown ∩
    • Has a maximum point (vertex is the highest point)
    • Increases before the vertex, decreases after the vertex

Vertex Formula

For any quadratic function y=ax2+bx+cy = ax^2 + bx + c, the x-coordinate of the vertex is: \(x = \frac{-b}{2a}\)

And that's it! We have all the information to determine when it is an increasing function and when it is decreasing - the vertex of the parabola is the highest or lowest point according to the concerning parabola. All that remains for us to do is, draw a sketch to see in it the intervals of increase and decrease clearly.

click here to read more about the vertex of the parabola!


Let's see an example:
When we see that the XX of the vertex is 55
and we conclude that the parabola has a happy face, we will make a small drawing:

It can be clearly seen that the function is increasing when X>5  X>5   and decreasing when X<5X<5 , therefore:
Interval of increase: X>5X>5
Interval of decrease: X<5 X<5 

Additional Example

Let's try an example with a "sad face" (downward-opening) parabola: Consider y=−x2+6x−5y = -x^2 + 6x - 5

Step 1: Find the vertex

  • Here a=−1a = -1, b=6b = 6, so XX of vertex = −62(−1)=3\frac{-6}{2(-1)} = 3

Step 2: Determine the face

  • Since a=−1<0a = -1 < 0, this is a "sad face" (maximum point)

Step 3: Identify intervals

  • Interval of increase: X<3X < 3 (before the vertex)
  • Interval of decrease: X>3X > 3 (after the vertex)

Examples and exercises with solutions on intervals of increase and decrease of a parabola

Exercise #1

Note that the graph of the function intersects the x-axis at points A and B

Moreover the vertex of the parabola is marked at point C

Identify the segment below where the function increases:

BBBAAACCC

Video Solution

Step-by-Step Solution

From the graph we can see that the parabola is a smiling parabola,

meaning that its extreme point is a minimum point.

If we describe it in words, until the extreme point the function decreases,

after the extreme point it increases.

Since we measure the progress using X,

we can say that the function increases whenever X is greater than point C, the extreme point.

Mathematically we can write:

X>C

As we already said, as long as X is greater than C, the function increases.

Answer

x>C x > C

Exercise #2

Find the domain of decrease of the function:

y=−x2+2x+35 y=-x^2+2x+35

Video Solution

Step-by-Step Solution

To determine the domain over which the quadratic function y=−x2+2x+35 y = -x^2 + 2x + 35 is decreasing, we proceed by identifying the vertex of the parabola.

Given the form y=ax2+bx+c y = ax^2 + bx + c , we have a=−1 a = -1 , b=2 b = 2 , and c=35 c = 35 . The x-coordinate of the vertex can be found using the formula:

x=−b2a x = -\frac{b}{2a}

Substituting b=2 b = 2 and a=−1 a = -1 into the formula, we calculate:

x=−22×(−1)=−2−2=1 x = -\frac{2}{2 \times (-1)} = -\frac{2}{-2} = 1

The vertex of the parabola occurs at x=1 x = 1 . Since the function is a downward-opening parabola (as indicated by the negative coefficient of −x2 -x^2 ), the function decreases for all x x values greater than the x-coordinate of the vertex.

Therefore, the domain of decrease for the function is x>1 x > 1 .

This matches the answer choice:

x>1 x > 1

Answer

x>1 x > 1

Exercise #3

Find the domain of increase of the function:

y=−x2+2x+35 y=-x^2+2x+35

Video Solution

Step-by-Step Solution

To find the domain of increase for the function y=−x2+2x+35 y = -x^2 + 2x + 35 , let's determine the vertex first.

  • Step 1: Identify coefficients in the quadratic equation. Here, a=−1 a = -1 , b=2 b = 2 , and c=35 c = 35 .
  • Step 2: Use the vertex formula x=−b2a x = -\frac{b}{2a} to find the x-coordinate of the vertex.

Plug in the values for b b and a a :

x=−22×−1=−2−2=1 x = -\frac{2}{2 \times -1} = -\frac{2}{-2} = 1

The x-coordinate of the vertex is x=1 x = 1 .

Since the coefficient a a is negative, this means the parabola opens downwards. A parabola opening downward will increase until it reaches the vertex, then start decreasing.

Therefore, the domain on which the function is increasing is x<1 x < 1 .

Therefore, the solution to the problem is x<1 x < 1 .

Answer

x<1 x < 1

Exercise #4

Find the domain of increase of the function:

y=3x2−6x+4 y=3x^2-6x+4

Video Solution

Step-by-Step Solution

The function given is y=3x2−6x+4 y = 3x^2 - 6x + 4 .

First, let's find the derivative of the function, which will help us determine the intervals of increase.

The derivative is given by f′(x)=ddx(3x2−6x+4)=6x−6 f'(x) = \frac{d}{dx}(3x^2 - 6x + 4) = 6x - 6 .

Next, find where the derivative is zero to locate critical points. Solve 6x−6=0 6x - 6 = 0 to get:

6x=6 6x = 6
x=1 x = 1

The critical point is x=1 x = 1 . This is where the function changes from decreasing to increasing since quadratic functions have one axis of symmetry and a>0 a > 0 : indicating a parabola opening upwards.

To determine the interval of increase, analyze the sign of f′(x) f'(x) :

  • For x>1 x > 1 , 6x−6>0 6x - 6 > 0 which implies f′(x)>0 f'(x) > 0 , so the function is increasing.
  • For x<1 x < 1 , 6x−6<0 6x - 6 < 0 which implies f′(x)<0 f'(x) < 0 , so the function is decreasing.

Thus, the domain of increase for the function is when x>1 x > 1 .

The correct answer is therefore x>1 x > 1 .

Answer

x>1 x > 1

Exercise #5

Note that the graph of the function below does not intersect the x-axis

Moreover the parabola's vertex is A

Identify the interval where the function is increasing:

XXXAAA

Video Solution

Answer

x<A x < A

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Test your knowledge
Question 1

Based on the data in the graph

Identify the domain with the decreasing function:

777111444

Question 2

Based on the data in the graph

Identify the domain where the function is increasing:

555

Question 3

Note that the graph of the function intersects the x-axis at points A and B

Moreover the vertex of the parabola is marked at point C

Identify the segment below where the function increases:

BBBAAACCC

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