Question

which translation maps the vertex of the graph of the function f(x) = x2 onto the vertex of the function g(x) = x2 – 10x 2? right 5, down 23 left 5, down 23 right 5, up 27 left 5, up 27

Answer 1

Answer:

$5$ units to the right and $23$ units down

Step-by-step explanation:

we have

$f(x)=x^{2}$

This is a vertical parabola open upward with vertex at point $(0,0)$

$g(x)=x^{2} -10x+2$

Step 1

Convert the function g(x) into vertex form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

$g(x)-2=x^{2} -10x$

Complete the square. Remember to balance the equation by adding the same constants to each side

$g(x)-2+25=x^{2} -10x+25$

$g(x)+23=x^{2} -10x+25$

Rewrite as perfect squares

$g(x)+23=(x-5)^{2}$

$g(x)=(x-5)^{2}-23$

The function g(x) is a vertical parabola with the vertex at point $(5,-23)$

Step 2

Find the rule of the translation

$(0,0)-------> (5,-23)$

$(x,y)-------> (x+5,y-23)$

That means

The translation is $5$ units to the right and $23$ units down

Answer 2

Vertex form of the function g(x) is:
g (x) = x² - 10 x + 2 = x² - 10 x + 25 - 25 + 2=
= ( x - 5 )² - 23
h = 5,  k = - 23
Answer: A) Right 5, down 23.