Question

the curve y=x^2-x+1 is translated by (0,3)
find and simplify the equation of the translated curve

Answer 1

1) If the graph of  $y=f(x)$  is translated $a$ units horizontally and $b$ units vertically, then the equation of the translated graph is  $y-b=f(x-a).$

$$y-0=(x-2)^2+3(x-2)+4$$ $$y=x^2-4x+4+3x-6+4$$ $$y=x^2-x+2$$

2) We start with the graph of $y=f(x).$ The graph of $g(x)=f(-x)$ is the graph of $y=f(x)$ reflected in the $y$ -axis . How points in graph of $f(x)$ become points of new graph $g(x)$ ?

$$(a,b)\to(-a,b)$$

Visual effect: flip over the $y$ -axis

The graph of $h(x)=3g(x)=3f(-x)$ is the graph of $y=g(x)$ stretched vertically by $3$ units.

How points in graph of $g(x)$ become points of new graph $h(x)$?

$$(-a,b)\to(-a,3b)$$

Visual effect: stretch vertically by $3.$

How points in graph of $y=f(x)$ become points of new graph $y=3f(-x)$?

$$(a,b)\to(-a,3b)$$