Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quadratic Equation Straight Line

  1. Oct 24, 2004 #1
    Hi,

    Please Can You Help Me To Understand How To Find The Vertex Of A Quadratic Equation (x,y Coordinates) Just By Looking At The Equation ?

    Eg What Is The Vertex Of Y=3x^2+4x+7 ?


    __________________________________________________


    Also, How Do I Transform A Straight Line Graph Left Or Right On The x Axis Without Moving The Line Up/down Along The Y ?


    Thanx


    Roger
     
  2. jcsd
  3. Oct 24, 2004 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Complete the square.

    If y= 3x2+4x+ 7, rewrite it as y= 3(x2+ (4/3)x)+ 7
    A "perfect square" is of the form (x+ a)2= x2+ ax+ a2. Compare that to x2+ (4/3)x. The coefficient of x is 4/3. 2a= 4/3 if a= 2/3 and, in that case, a2= 4/9. If we add 4/9 then x2+ (4/3)x+ 4/9= (x+ 2/3)2, a perfect square.

    Of course, we can't JUST add 4/9, that would change the value. We can, however both add and subtract 4/9:

    y= 3(x2+ (4/3)x+ 4/9- 4/9)+ 7 and now take the "-4/9" out of the parentheses:
    = 3(x2+ (4/3)x+ 4/9)- 4/3+ 7
    = 3(x+ 2/3)2+ 17/3

    If x= -2/3, then x+ 2/3= 0 so y= 17/3. If x is ANY OTHER number, x+ 2/3 is NOT 0 so the square is positive and y is 17/3 PLUS some number: larger than 17/3.

    The lowest point on the parabola, the vertex, is (-2/3, 17/3).

    To "transform a straight line left or right on the x-axis", in other words, translate left or right, ADD OR SUBTRACT SOMETHING FROM x.

    If the equation of the line is y= 5x+ 3, then y= 5(x- 2)+ 3 = 5x- 10+ 3= 5x- 7 has graph a line parallel to y= 5x+ 3 but shifted right 2 places: y= 3 in 5x+ 3 when x= 0 but in y= 5x- 7 when x= 2.

    The graph of any function, y= f(x), is shifted to the right a places is x is replaced by x-a: y= f(x-a).
     
    Last edited: Oct 24, 2004
  4. Oct 24, 2004 #3

    thanx for the help.

    please can you go over finding the vertex..


    I thought the 17/3 bit is simply the y intercept ?



    Also for a straight line graph, is that to say you cant move the line left or right withought affecting the y intercept ?

    roger
     
  5. Oct 24, 2004 #4

    robphy

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Here's a calculus-based approach:

    y= 3x2+4x+ 7
    has its extremum when dy/dx=0

    so, 0=(dy/dx)at extremum=6xat extremum+4

    thus, xat extremum=-2/3
    and yat extremum=3(-2/3)2+4(-2/3)+7

    Here's a variant, inspired by kinematics:

    y=(1/2)at2+v0t+y0
    v=at+v0

    0=v at the maximum height. So, tat extremum=-v0/a.

    Thus, think of the original equation as
    y=(1/2) 6x2+4x+ 7
    so, xat extremum=-(4)/(6).
     
  6. Oct 24, 2004 #5
    y=ax²+bx+c
    xextremum=-b/2a

    Proof:
    dy/dx=2ax+b
    At the minima and maxima points of the curve, dy/dx=0 => 2ax+b=0 => x=-b/2a

    This doesn't work all the time though.
     
  7. Oct 24, 2004 #6

    robphy

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Ah, yes, of course.
    The x-coordinate of the vertex of y=ax2+bx+c is -b/2a,
    which can be interpreted as the average of the roots of "0=ax2+bx+c".
     
  8. Oct 24, 2004 #7

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    No, the orginal equation was y= 3x2+ 4x+ 7= 0. The y intercept is, by definition, the y coordinate of the point where the graph crosses the y-axis. Since every point on the y-axis has x-coordinate 0, the y intercept is
    y= 3(0)2+ 4(0)+ 7= 7.



    That's exactly what it says. The line y= x+ 1 has y intercept 1. If fact, since it has slope 1, the distance from the x intercept to the origin is exactly the same as the distance from the origin to the y intercept: 1. If I move the line parallel to itself, say to the left 2 spaces, it still has slope 1 (that depends only on the angle) so distance from the x intercept to the origin must still be the same as the distance from the origin to the y intercept.
    Moving to the left 2 spaces the x intercept is now (-3,0) so the y intercept is (0,3).
    Translating a line to the left or right keeps the slope the same, not the intercepts.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Quadratic Equation Straight Line
  1. Straight line? (Replies: 35)

  2. Euclidstraight line (Replies: 4)

Loading...