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skelet666 [1.2K]
3 years ago
8

What steps transform the graph of y=x^2 to y= -(x+3)^2+5

Mathematics
1 answer:
Lady bird [3.3K]3 years ago
4 0
\bf ~~~~~~~~~~~~\textit{function transformations}
\\\\\\
% templates
f(x)=  A(  Bx+  C)+  D
\\\\
~~~~y=  A(  Bx+  C)+  D
\\\\
f(x)=  A\sqrt{  Bx+  C}+  D
\\\\
f(x)=  A(\mathbb{R})^{  Bx+  C}+  D
\\\\
f(x)=  A sin\left( B x+  C  \right)+  D
\\\\
--------------------

\bf \bullet \textit{ stretches or shrinks horizontally by  }   A\cdot   B\\\\
\bullet \textit{ flips it upside-down if }  A\textit{ is negative}\\
~~~~~~\textit{reflection over the x-axis}
\\\\
\bullet \textit{ flips it sideways if }  B\textit{ is negative}

\bf ~~~~~~\textit{reflection over the y-axis}
\\\\
\bullet \textit{ horizontal shift by }\frac{  C}{  B}\\
~~~~~~if\ \frac{  C}{  B}\textit{ is negative, to the right}\\\\
~~~~~~if\ \frac{  C}{  B}\textit{ is positive, to the left}\\\\
\bullet \textit{ vertical shift by }  D\\
~~~~~~if\   D\textit{ is negative, downwards}\\\\
~~~~~~if\   D\textit{ is positive, upwards}\\\\
\bullet \textit{ period of }\frac{2\pi }{  B}

now, with that template in mind, let's check this one

\bf y=x^2\implies y=\stackrel{A}{1}(\stackrel{B}{1}x\stackrel{C}{+0})^2\stackrel{D}{+0}
\\\\\\
y=-(x+3)^2+5\implies y=\stackrel{A}{-1}(\stackrel{B}{1}x\stackrel{C}{+3})^2\stackrel{D}{+5}

A = -1, reflection over the x-axis.

B = 1, C = +3, C/B = +3/1 = +3, horizontal shift to the left of 3 units.

D = 5, vertical shift up of 5 units.
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