The axis of symmetry of a parabola is halfway between the two x-intercepts.
Function: P(x)=x(10-x) Two numbers that give maximum product: 5,5
Practice makes perfect
Let x be one of the numbers.
If the sum of the two numbers is 10, then the other number is 10-x.
Function Representing the Product
Let's write a function that represents the product of these two numbers.
P(x)=x(10-x)
Now we investigate the graph that shows the product of the two numbers.
We will use the information in the question to find the x-intercepts, the axis of symmetry, and the vertex of this graph.
x-intercepts
The x-intercepts correspond to numbers for which the product is 0.
The product of two numbers can only be 0 when one of the numbers is 0.
Since the sum of the two numbers is 10, if one is 0 then the other is 10.
x-intercepts: (0,0)and(10,0)
Axis of Symmetry
The graph is symmetric to the axis of symmetry, so the axis of symmetry is a vertical line halfway between 0 and 10, the two x-intercepts.
Axis of symmetry: x=5
Vertex
The vertex is on the axis of symmetry, so it's first coordinate is 5.
If one of the numbers is 5 and the sum of the two numbers is 10, then the other number is 10-5=5.
The product of these two numbers is 5* 5=25.
Vertex: (5,25)
The two numbers that give the maximum product of 25 is 5 and 5.
