Answer
The required parabola is shown below.
Work Step by Step
The quadratic function can be expressed as:
$\begin{align}
& f\left( x \right)={{x}^{2}}+3x-10 \\
& ={{x}^{2}}+3x+{{\left( \frac{3}{2} \right)}^{2}}-10-\frac{9}{4} \\
& ={{\left( x+\frac{3}{2} \right)}^{2}}-\frac{49}{4}
\end{align}$
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 $f\left( x \right)={{\left( x-\left( -\frac{3}{2} \right) \right)}^{2}}+\left( -\frac{49}{4} \right)$ is $\left( h,k \right)=\left( -\frac{3}{2},-\frac{49}{4} \right)$.
Step 2:
The standard parabola is also symmetric with respect to the line $x=h$. So, the provided parabola is symmetric to $x=-\frac{3}{2}$.
Step 3:
If $a>0$ , the parabola opens upward and if $a<0$ then the parabola opens downward. 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 graph opens upward as $a>0$.
Step 4:
So, we see that the above steps lead to the parabola that is open upwards, has a vertex at $\left( -\frac{3}{2},-\frac{49}{4} \right)$ and intercepts as $\left( -5,0 \right)\text{,}\left( 2,0 \right)\text{ and }\left( 0,-10 \right)$. Thus, the required parabola is shown above.
