Translation of a Conic Section

Key Questions

  • When translating a graph to the left or to the right, it means moving the entire graph to the left or to the right of its current location, which is usually relative to #(0, 0 )#

    For example, let's say you have a circle centered at #(0, 0)# with radius 2. Its standard equation would be

    #x^2 + y^2 = 4#

    Now, let's say we translate the circle 5 units to the left. Your circle will now be centered at (-5, 0) with the radius still equal to 2. Its new standard equation would be

    #(x + 5)^2 + y^2 = 0#


    This is also true for linear equations.
    For example, let's say you have a line with slope 1 with the x-intercept equal to 0. Its equation will be

    #y = x#

    When we translate the line 3 units to the right, its slope will remain the same, but its x-intercept will now be 3. Its new equation will be

    #y = x - 5#

  • Translations are simply horizontal and/or vertical shifts. So, #y = sqrtx - 2# is shifted 2 units down from #y = sqrtx#, while #y = sqrt(x-2)# is shifted 2 units right of #y = sqrtx#.

    #y = sqrtx#:
    graph{sqrtx [-10, 10, -5, 5]}

    #y = sqrtx - 2#:
    graph{sqrtx - 2 [-10, 10, -5, 5]}

    #y = sqrt(x - 2)#:
    graph{sqrt(x-2) [-10, 10, -5, 5]}

    #y = sqrt(x - 2) - 2#:
    graph{sqrt(x-2) - 2 [-10, 10, -5, 5]}

  • You can translate any function, #y=f(x)# using:

    #y=f(x-h)+k# or
    #y-k=f(x-h)#

    If #h# is positive, the graph will translate to the right.
    If #h# is negative, the graph will translate to the left.
    If #k# is positive, the graph will translate up.
    If #k# is negative, the graph will translate down.

    Here is an example:

    #f(x)=x^2+2x#

    If we want to translate this up 4 units, then we have:

    #y=f(x)+4#
    #y=x^2+2x+4#
    enter image source here

    If we have a different example:

    #f(x)=1/x#
    #y=f(x)+4#
    #y=1/x+4#
    enter image source here
    It still works!

Questions