# Chapter 2 - Section 2.2 - Quadratic Functions - Exercise Set - Page 330: 25

The required parabola is shown below.

#### Work Step by Step

Use the steps shown below to determine the graph of the quadratic equation. Step 1: The quadratic function can be written as $f\left( x \right)=a{{\left( x-h \right)}^{2}}+k$ corresponding to which the graph is a parabola whose vertex is at $\left( h,k \right)$. Thus, the vertex of the parabola $f\left( x \right)=-{{\left( x-1 \right)}^{2}}+4$ is $\left( h,k \right)=\left( 1,4 \right)$. Step 2: The parabola is also symmetric with respect to the line $x=h$. So, the provided parabola is symmetric to $x=1$. Step 3: If $a>0$ , the parabola opens upwards and if $a<0$ then the parabola opens downwards. Also, if $\left| a \right|$ is small, the parabola opens more flatly than if $\left| a \right|$ is large. And, from the provided equation of the function, it is observed that the graph opens downwards as $a<0$. Step 4: So, we see that the above steps lead to the parabola that opens upwards and has vertex at $\left( 1,4 \right)$. Therefore, the required parabola is shown above.

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