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Q. 50

Expert-verifiedFound in: Page 156

Book edition
6th

Author(s)
Sullivan

Pages
1200 pages

ISBN
9780321795465

Determine the quadratic function whose graph is given.

The quadratic function whose graph is given is $f\left(x\right)=-(x-2{)}^{2}+3$ or $f\left(x\right)=-{x}^{2}+4x-1$.

The given graph is:

The vertex is $(2,3)$, so $h=2$ and $k=3$. Substitute these values into equation $f\left(x\right)=a(x-h{)}^{2}+k$.

$f\left(x\right)=a(x-2{)}^{2}+3$

To determine the value of $a$, we use the fact that $f\left(0\right)=-1$ (the *y*-intercept).

$\begin{array}{rcl}-1& =& a(0-2{)}^{2}+3\\ -1-3& =& 4a\\ -4& =& 4a\\ -1& =& a\end{array}$

Substitute $a=-1$ in $f\left(x\right)=a(x-2{)}^{2}+3$.

$f\left(x\right)=-1(x-2{)}^{2}+3\phantom{\rule{0ex}{0ex}}f(x)=-(x-2{)}^{2}+3\phantom{\rule{0ex}{0ex}}f\left(x\right)=-{x}^{2}+4x-4+3\phantom{\rule{0ex}{0ex}}f\left(x\right)=-{x}^{2}+4x-1$

The quadratic function whose graph is given is $f\left(x\right)=-(x-2{)}^{2}+3$ or $f\left(x\right)=-{x}^{2}+4x-1$.

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