Sketching quadratic graphs are drawn based on $y=x^2$ graph for transforming and translating.

## Question 1

$f(x) = (x-3)^2$ is drawn and sketch the following graphs by transforming.

(a)   $y = f(x)+2$; Transforming upwards by $2$ units

(b)   $y=f(x)-3$; Transforming dowanwards by $3$ units

(c)   $y=-f(x)$; Rotating by $x$-axis

(d)   $y=f(-x)$; Rotating by $y$-axis

(e)   $y=f(x+2)$; Transforming to the left by $2$ units

(f)   $y=f(x-1)$; Transforming to the right by $1$ unit

(g)   $y=f(x+4)-5$; Transforming to the left by $4$ units and downwards by $5$ units

(h)   $y=-f(x+2)$; Rotating to $x$-axis, then transforming to the left by $2$ units

(i)   $y=2f(x)$

(j)   $\displaystyle y=\frac{1}{2}f(x)$

(k)   $y=-4f(x)$

(l)   $y=3f(x+5)$

(m)   $y=-f(x+6)-3$

(n)   $y=2f(x)-4$

(o)   $y=-2f(x+3)+1$

(p)   $y=2f(-x)-2$

## Question 2

(a)   $y=(x-1)^2$

(b)   $y=(x+2)^2$

(c)   $y=(x+5)^2-3$

(d)   $y=(x-4)^2+3$

(e)   $y=-(x+2)^2$

(f)   $y=-(x-3)^2$

(g)   $y=-(x-3)^2+2$

(h)   $y=-(x+4)^2-2$ ## Mastering Integration by Parts: The Ultimate Guide

Welcome to the ultimate guide on mastering integration by parts. If you’re a student of calculus, you’ve likely encountered integration problems that seem insurmountable. That’s…