I don't know how to do this, please help!

Question

How is the location of the vertex related to the x-intercepts?

 

n. Make an ordered list of steps necessary to find the vertex once you have the x-intercepts

 

o. Give an example of a parabola that has no x-intercepts

 

p. Define the concepts of "maxima" and "minima"

 

(please show work) THNX!

Answer 1

(n)  if the x-intercepts are (a,0) and (b,0), where a < b, then from the symmetry properties of the parabola, the x-coordinate of the vertex (usually denoted as "h") would be given as h = (a + b)/2, and the y-coordinate of the vertex (usually denoted as "k") would be given as k = f(k), where the equation of the parabola is given in "vertex form", i.e.,

y = f(x) = A(x - h)^2 + k

 

(o)  a parabola that has no x-intercepts would be of the form: y = f(x) = A(x - h)^2 + k, where either

      (a) A >0, and k > 0, or (b) A < 0, and k < 0

 

      examples:   (a) y = f(x) = 2(x - 3)^2 + 4           (b) y = f(x) = -2(x - 3)^2 - 4

 

(p)  f(x) is "max" at x = a if (i) f(x) < f(a), for x < a; and (ii) f(x) < f(a) for x > a (draw a picture)

 

      f(x) is "min" at x = a if (i) f(x) > f(a) for x < a; and (ii) f(x) > f(a) for x > a  (draw a picture)

 

Answer 2

A parabola is a symmetrical curve.  The x-coordinate of the vertex serves as the axis of symmetry of the parabola.  You can say the axis of symmetry divides the parabola into two equal halves.

 

Also, the parabola can only intercept the x-axis at most 2 times.  Therefore, the vertex is the average between the 2 x-intercepts.

 

 

In terms of ordered list of steps:

 

Use the formula to find the x-coordinate of the vertex.

 

x = -b / 2a

 

This formula comes from taking the average of the x-intercepts, as I mentioned before.  If you recall the quadratic formula, the solutions come in conjugates.

 

x₁ = (-b - √(b² - 4ac)) / 2a

x₂ = (-b + √(b² - 4ac)) / 2a

 

where  a, b, and c are constants in the quadratic equation

 

ax² + bx + c = 0

 

 

Taking the average of x₁ and x₂ gives us

 

x_ave = (1 / 2)(-2b / 2a)

      = (1 / 2)(-b / a)

      = -b / (2a)

 

Then, you will evaluate the expression of the parabola at x_ave to find the y-coordinate of the vertex.

 

 

An example of a parabola with no x-intercepts

 

y = ax² + bx + c

 

Under these conditions:

 

1)   a>0   ,   b=0   ,     c>0

2)   a<0   ,   b=0   ,     c<0

 

A maxima is the highest value in the parabola in a given domain.  A minima is the lowest value in that parabola in a given domain.