What is the the vertex of y = 1/2(x+1)(x-5) ?

3 Answers
Dec 13, 2015

y= 1/2 (x-color(red)(2))^2 color(blue)(-9/2)

vertex: (2, -9/2)

Explanation:

Note:

Vertex form f(x) = a(x-h)^2+k
h= x_(vertex) = -b/(2a) " " " " ; k= y_(vertex)= f(-b/(2a))

Given:

y= 1/2 (x+1)(x-5)

Multiply the expression or FOIL

y = 1/2 (x^2 -5x+x-5)

y = 1/2(x^2 -4x-5)

y= 1/2x^2 -2x -5/2

a = 1/2;" " b= -2;" " " c= -5/2

color(red)(h= x_(vertex)) = (-(-2))/(2*1/2) =color(red) 2
color(blue)(k= y_(vertex)) = f(2) = 1/2(2)^2 -2(2) -5/2

=> 2-4 -5/2 => -2 -5/2 => color(blue)(-9/2

The vertex form is

y= 1/2 (x-color(red)(2))^2 color(blue)(-9/2)

Dec 13, 2015

(2,-9/2)

Explanation:

First, find the expanded form of the quadratic.

y=1/2(x^2-4x-5)

y=1/2x^2-2x-5/2

Now, the vertex of a parabola can be found with the vertex formula:

(-b/(2a),f(-b/(2a)))

Where the form of a parabola is ax^2+bc+c.

Thus, a=1/2 and b=-2.

The x-coordinate is -(-2)/(2(1/2))=2.

The y-coordinate is f(2)=1/2(2+1)(2-5)=-9/2

Thus, the vertex of the parabola is (2,-9/2).

You can check the graph:
graph{1/2(x+1)(x-5) [-10, 10, -6, 5]}

Dec 14, 2015

color(blue)("A slightly quicker approach")

color(green)("It is not uncommon for there to be several ways of solving a problem!")

Explanation:

This is a quadratic thus of the hors shoe type shape.

That means that the vertex is 1/2 way between the x-intercepts.

The x-intercepts will occur when y=0

If y is 0 then the right side also = 0

The right side equals zero when (x+1)=0 " or " (x-5)=0

For (x+1)=0 -> x=-1
For(x-5)=0 -> x =+5

Half way is ((-1)+(+5))/2 = 4/2=2

Having found color(blue)(x_("vertex")=2) we then substitute in the original equation to find color(blue)(y_("vertex"))