which translation maps the vertex of the graph of the function $f(x) = x^2$ onto the vertex of the function…

which translation maps the vertex of the graph of the function $f(x) = x^2$ onto the vertex of the function $g(x) = -8x + x^2 + 7$?\n\\bigcirc left 4, down 9\n\\bigcirc left 4, up 23\n\\bigcirc right 4, down 9\n\\bigcirc right 4, up 23

which translation maps the vertex of the graph of the function $f(x) = x^2$ onto the vertex of the function $g(x) = -8x + x^2 + 7$?\n\\bigcirc left 4, down 9\n\\bigcirc left 4, up 23\n\\bigcirc right 4, down 9\n\\bigcirc right 4, up 23

Answer

Explanation:

Step1: Find vertex of ( f(x) )

The function ( f(x) = x^2 ) is a standard parabola. Its vertex form is ( y = a(x - h)^2 + k ), where ((h, k)) is the vertex. For ( f(x) = x^2 ), ( h = 0 ), ( k = 0 ), so vertex is ((0, 0)).

Step2: Rewrite ( g(x) ) in vertex form

Given ( g(x) = -8x + x^2 + 7 ), rearrange to ( g(x) = x^2 - 8x + 7 ). Complete the square for ( x^2 - 8x ):

  • Take coefficient of ( x ), which is (-8). Half of it is (-4), square of (-4) is (16).
  • So, ( g(x) = (x^2 - 8x + 16) - 16 + 7 = (x - 4)^2 - 9 ).

Step3: Find vertex of ( g(x) )

From vertex form ( g(x) = (x - 4)^2 - 9 ), the vertex ((h, k)) is ((4, -9)).

Step4: Determine translation

From ((0, 0)) to ((4, -9)):

  • Horizontal translation: ( 4 - 0 = 4 ) units to the right (since ( h ) increases from 0 to 4).
  • Vertical translation: ( -9 - 0 = -9 ) units, which means 9 units down (since ( k ) decreases from 0 to -9).

Answer:

right 4, down 9