Question

find g(x), where g(x) is the translation 10 units left and 4 units up of $f(x) = x^2$. write your answer in the form $a(x - h)^2 + k$, where a, h, and k are integers. $g(x) = \square$

Explanation:

Step1: Recall the translation rules for functions

For a function \( y = f(x) \), a horizontal translation \( h \) units left is given by \( y = f(x + h) \), and a vertical translation \( k \) units up is given by \( y = f(x) + k \). Also, the vertex form of a quadratic function is \( y=a(x - h)^2+k \), where \( (h,k) \) is the vertex.

Step2: Apply the horizontal translation

We need to translate \( f(x)=x^{2} \) 10 units left. Using the horizontal translation rule, replacing \( x \) with \( x + 10 \), we get \( f(x + 10)=(x + 10)^{2} \).

Step3: Apply the vertical translation

Now, we need to translate the function obtained in step 2, 4 units up. Using the vertical translation rule, we add 4 to the function. So \( g(x)=(x + 10)^{2}+4 \). We can rewrite \( (x + 10)^{2} \) as \( (x-(- 10))^{2} \), so in the form \( a(x - h)^{2}+k \), we have \( a = 1 \), \( h=- 10 \), and \( k = 4 \).

Answer:

\( 1(x - (-10))^{2}+4 \) or simply \( (x + 10)^{2}+4 \) (which is equivalent to \( 1(x-(-10))^{2}+4 \))