Question

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

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