Consider the quadratic function f ( x ) = ( x + 2 ) ( 5 x - 25 ) f(x)=(x+2)(5x-25).

Question

What are the x intercepts?

What is the y intercept?

What is the vertex point?

What was does the parabola open?

What is the equation for the line of symmetry?

Answer 1

f(x) = (x+2)(5x-25)

We can factor a 5 out of (5x-25) giving:

f(x) = 5(x+2)(x-5)

replace f(x) with y

 

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

 

a)  The x-intercepts are where y=0

5(x+2)(x-5) = 0 at x=-2 and x=5

x - intercepts are (-2,0) and (5,0)

 

b) y intercept is where x=0

    5(0+2)(0-5) = 5(2)(-5) = -50

    y intercept is at (0, -50)

 

c) Find the vertex

    the vertex of a quadratic y =ax²+bx+c

    is at (b/2a, f(-b/2a))

 

y = 5(x+2)(x-5)

y = 5(x²-3x-10)

y = 5x²-15x-50

a = 5, b=-15, c=-50

 

-b/2a = 15/10 = 3/2 or 1 1/2

 

f(3/2) = 5(3/2)²-15(3/2)-50

= 5(9/4) - 45/2 - 50

= -45/4 - 200/4

= -245/4

= -61 1/4

 

Vertex of the parabola is at

(3/2 , -245/4)

 

d)  Since the coefficient of the squared term in

     y= 5x²-15x-50 is positive, the parabola opens upward.

   

Note:  we can also tell this because the vertex is below the

          x-axis and there are real x-intercepts at (-2,0) and (5,0)

 

e)  The equation for the line of symmetry is the line through the

     x coordinate of the vertex because the square term is the x.

 

    x = 3/2 is the equation for the line of symmetry.

 

 

    

Answer 2

To find the x-intercepts, set f(x) equal to zero and solve for x.

 

(x + 2)(5x - 25) = 0

 

 

To find the y-intercept, set x=0 and solve for f(x).  That is f(0).

 

 

To find the vertex point:

 

First, find the x-coordinate of the vertex.  The x-coordinate of the vertex is the average of the x-intercepts. 

Then, once you found the x-coordinate of the vertex, evaluate f(x) at that x-coordinate to find the y-coordinate of the vertex.

 

 

When you expand f(x) using FOIL, the coefficient of the leading term is positive.  This means that the parabola opens up.  The parabola makes a smiley face shape because of the positive  sign of the leading coefficient.

 

 

The equation for the line of symmetry is the x-coordinate of the vertex.