I'm totally stuck!  I've tried working it but I seem to get stuck on the same

spot!  Here's the problem: 



For the function y = x2 - 4x - 5, perform the following tasks:



a) Put the function in the form y = a(x - h)2 + k.



b) What is the line of symmetry?

	

c) Graph the function using the equation in part a.  Explain why it is not

necessary to plot points to graph when using y = a (x – h)2 + k.



d) In your own words, describe how this graph compares to the graph of

   y = x2?



For "a", I got y=(x-2)2 - 14.  From there, I'm totally lost!  Can somebody

help!  (P.S.  I haven't taken algebra since high school and that was 15 yrs

ago!)  :-)_



Thanks!

 
This question is from textbook College Algebra


Found 2 solutions by  Earlsdon, AnlytcPhil:
Answer by Earlsdon(6294)   (Show Source): You can put this solution on YOUR website!
 



a) Convert toform.

 Add 5 to both sides.

 Complete the square in the x-terms by adding the square of half the x-coefficient to both sides.

 Factor the right side. Simplify the left side.

 Finally, subtract 9 from both sides.



Compare this with 

a = 1, h = 2, and k = -9



b) The line of symmetry (LOS) is the vertical line through the center of the vertex. The vertex is located at (h, k), so in this problem, the vertex is located at (2, -9).

The LOS is therefore the vertical line x = 2



c)& d) The graph of(in red).

The graph of (in green).



As you can see, the two parabolas have the same shape but the red one (your equation) is translated to the right by h units (h=2) and down k units (k = -9). So by first graphing the the parabola whose vertex lies at the origin, then translating (sliding) it so that the vertex is at the location defined by (h, k) in your equation, you really don't need to plot points. 

Answer by AnlytcPhil(1806)   (Show Source): You can put this solution on YOUR website!
 
I'm totally stuck! I've tried working it but I seem to get stuck on the
same spot! Here's the problem: 

For the function y = x2 - 4x - 5, perform the following tasks: 

a) Put the function in the form y = a(x - h)2 + k.

b) What is the line of symmetry?
 
c) Graph the function using the equation in part a.
   Explain why it is not necessary to plot points to graph when using
   y = a(x – h)2 + k.

d) In your own words, describe how this graph compares to the graph of 
   y = x2? 

For "a", I got y=(x-2)2 - 14.

Sorry, that's wrong.  Let's go through it.

y = x2 - 4x - 5

Put a 1 for the coefficient of x2

y = 1x2 - 4x - 5

1. Factor the coefficient of x2 out of the first two terms.

This is a = 1.  So we factor "a" out of the first two terms:

In this case we factor 1 out of the first two terms

y = 1(x2 - 4x) - 5

2.  Now to the side, we multiply the coefficient of x  by 1/2,
(always 1/2).

In this case we multiply -4 by 1/2, getting -2

3.  Now, also to the side, we square this. 

In this case we square -2 and get +4.

4.  Now we 
    A. Add this number inside the parentheses
    B. Mentally multiply it by "a"
    C. Subtract this from the right side.
  
In this case we
(A) add +4 to the right side of the parentheses, 
(B) multiply +4 by a=1, getting +4 and 
(C) subtract 4 from the right side:

y = 1(x2 - 4x + 4) - 5 - 4

5.  Now we factor the expression in parentheses and combine 
    the terms after the parentheses.

In this case we factor x2 - 4x + 4 as (x-2)(x-2) or (x-2)2 and combine
    -5-4 as -9

y = 1(x - 2)2 - 9

-----------------

b) What is the line of symmetry?

For the equation

y = a(x - h)2 + k

the line of symmetry is always the vertical line whose equation is x = h
which bisects the parabola

In this case 

y = 1(x - 2)2 - 9

a = 1, h = 2, k = -9

the axis of symmetry is the vertical line whose equation is x = 2

c) Graph the function using the equation in part a.

 

Explain why it is not necessary to plot points to graph when using
y = a(x – h)2 + k.

A. plot the vertex which is (h,k)
B. plot the two points (h-1,k+a) and (h+1,k+a)

In this case we plot (A) the vertex as (h,k) = (2,-9) and
(B) the two points (h-1,k+a) = (2-1, -9+1) = (1,-8) and 
(h+1,k+a) = (2+1, -9+1) = (3,-8)

d) In your own words, describe how this graph compares to the 
   graph of y = x2?

(A) In y=x2, x has been replaced by (x-2) which means 
    that the graph has been shifted to the right by 2 units

    and

(B) 9 has been subtracted from the right side which means that 
    the graph has been shifted down by 9 units. 

Edwin
AnlytcPhil@aol.com


  

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