For AI agents: use /llms.txt for the Nakafa content index.
Ever seen a garden fence? Sometimes, we have a limited length of fence, but we want to make the garden as big as possible. Well, quadratic functions can help us find the garden dimensions that give the largest area! Magic, right?
A quadratic function looks like a smile ( if ) or a frown ( if ).
If we want to find the biggest value (maximum), we use the frown shape, so the value of is negative ().
The general form is:
Here, , , and are numbers, and cannot be zero (because if it's zero, it becomes linear).
The largest value is at the very top point of the frowning graph. This point is called the vertex.
To find the position of the vertex (the value), we use the formula:
After getting , we plug it back into the quadratic function to get the largest value (the or value):
Or you can use this shortcut formula:
where (D is called the discriminant).
Let's say a farmer has of fence. He wants to build a rectangular chicken coop. What are the dimensions of the coop so that the area is maximized?
Give Names: Let the length be and the width be .
Fence Length Relation: The fence is the perimeter.
Simplify (divide everything by ):
Meaning:
Area Formula: The area of the coop is . Substitute :
Quadratic Function Form: Arrange it neatly:
This is a quadratic function with , , and . Since is negative, the graph is a frown, so there is a maximum value.
Find Maximum Width (l): Use the formula (but replace with ):
Find Maximum Area (A): Plug into the area formula :
Conclusion: For the maximum coop area (), the width . The length is . The dimensions must be (it turns out to be a square!).
Finding the biggest profit.
Example:
A toy store sells dolls. If they sell dolls, the profit (in thousands of rupiah) is . How many dolls should be sold to maximize profit?
Calculating the highest point of a thrown object's jump.
Example:
A toy rocket flies! Its height after is . When is the rocket highest and what is its maximum height?
A rectangle has a perimeter of . Determine its length and width to maximize its area, and calculate the maximum area!
Let length be , width be .
Perimeter:
Area:
Area Function:
().
Width for max area ():
Length at max area ():
Maximum area ():
So, for the maximum area (), the dimensions must be x (a square again!).
So, the width must be for the maximum area.
The largest area is .
