I do not understand at all, please help

Question

How is graphing form useful for indentifying important features of a parabola?

 

q. Create an ordered list of the steps necessary to "complete the square"

 

r. Explain how you can find the vertex of a parabola using vertex form.

 

s. Describe how you would find the x-intercept using graphing form -- use this example to show how solving can show that parabolas sometimes don't have x-intercepts.

 

(Please show work) THANK YOU.!

Answer 1

(q)  if you start with the Quadratic Equation, ax^2 + bx  + c = 0, then write the following to complete the square (the

      following is essentially the derivation of the Quadratic Formula):

 

     (i)    subtract "c" from both sides of the equation:     ax^2 + bx = -c

 

     (ii)  divide both sides of the equation by "a" (so that the coefficient of x^2 is one):   x^2 + (b/a)x = -(c/a)

 

     (iii)  complete the square, i.e., add [b/(2a)]^2 to each side of the equation:  

 

                   x^2 + (b/a)x + [b/(2a)]^2 = -(c/a) + [b/(2a)]^2

 

      (iv)  the left side of the equation is now a perfect square;  

             also, re-write the right side of the equation so as to get a common denominator:

 

             [x + b/(2a)]^2 = (b^2 - 4ac)/(4a^2)

 

      (v)  use Square Root Property, i.e., take square roots on both sides of the equation:

            (assume a > 0;  this is actually not necessary, but can always be done, i.e., make a > 0, and doing so makes

             things simpler)

 

             x + b/(2a) = +/- sqrt(b^2 - 4ac)](2a)

 

 

     (vi)   subtract b/(2a) from each side of the equation (to obtain Quadratic Formula):

 

             x = [-b +/-sqrt(b^2 - 4ac)]/(2a)

 

     for any particular numerical values of a, b, c, just repeat above steps with those numerical values.

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(r)  if equation of parabola is in "vertex" form, y = a(x - h)^2 + k, then coordinates of vertex are (h,k);  draw a picture

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(s)  if equation of parabola is in "vertex" form, y = a(x - h)^2 + k, then to find x-intercepts, where y = 0, then solve the following equation:

     

                            a(x - h)^2 + k = 0 ---> a(x - h)^2 = - k ---> (x - h)^2 = -(k/a) ---> (x - h) = +/-sqrt[-(k/a)] --->

   

                            x = h +/-sqrt[-(k/a)];  note: if either (i) k and a are each positive, or (ii) k and a are each negative,

 

                                                                    then the quantity under the square root sign will be negative, and 

 

                                                                    sqrt(negative) = imaginary ---> no x-intercepts

 

     

 

see answer to previous question; 

 

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Answer 2

q)

 

Follow this link for a description on how to solve complete the square.

https://www.wyzant.com/resources/files/467144/deriving_the_quadratic_formula

 

r)

 

The vertex form is

 

y = a(x - h)² + k

 

where:

a is the leading coefficient

(h, k) is the vertex

 

 

s)

 

At which value is y=0.  Or find the points where the curve crosses the x-axis.