Q.Write the steps to obtain the graph of the function $y=3(x-1)^2+5$ from the graph $y=x^2$.

Rather than plotting a complicated curve point by point, recognise it as a reflection, translation, or dilation of an already-known simpler curve. Reflection and translation keep the graph congruent (same size and shape); dilation changes the shape by stretching/compressing.

Reflection (mirror image about a line):

Translation (slide without changing shape), for $c,d>0$:

Dilation (stretch/compress): multiplying $f$ by a constant $k>0$ scales the graph vertically -- $k>1$ stretches it away from the $x$-axis, $0<k<1$ compresses it toward the $x$-axis. Replacing $x$ by $kx$ inside $f$ instead rescales horizontally (e.g. $y=\sin2x$ compresses $y=\sin x$ horizontally by factor 2, halving the distance between successive $x$-intercepts while keeping the same $\pm1$ amplitude). …