Q44. - Algebra 2

Question

The shape of each arch supporting the Exchange House can be modeled by $h\left(x\right)=-0.025x^2+2x$, where $h\left(x\right)$ represents the height of the arch and x represents the horizontal distance from one end of the base in meters.

 

Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of $h\left(x\right)$.

Step-by-Step Solution

Answer

The equation of the axis of symmetry is $x=40$ and the coordinates of the vertex of the graph is $\left(40,40\right)$.

Step 1. Given Information.

Given that the shape of each arch supporting the Exchange House can be modeled by $h\left(x\right)=-0.025x^2+2x$, where $h\left(x\right)$ represents the height of the arch and x represents the horizontal distance from one end of the base in meters.

 

The equation of the axis of symmetry, and the coordinates of the vertex of the graph of $h\left(x\right)$ are to be determined.

Step 2. Explanation .

For an equation of the form $f\left(x\right)=a{x}^2+bx+c$, the axis of symmetry is given by $x=-\frac{b}{2a}$

Here for the given equation, $a=-0.025,b=2$

Plugging the values in the equation:

 $\begin{array}{l}x=-\frac{b}{2a}\\ x=-\frac{2}{2\left(-0.025\right)}\\ x=\frac{200}{5}\\ x=40\end{array}$

Hence the axis of symmetry is $x=40$.

 

The x-coordinate of the vertex is same as the axis of symmetry.

So the y-coordinate of the vertex can be obtained by plugging the value in the equation.

Plugging $x=40$ in the equation:

 $\begin{array}{l}h\left(x\right)=-0.025x^2+2x\\ h\left(40\right)=-0.025{\left(40\right)}^2+2\left(40\right)\\ h\left(40\right)=-0.025\left(1600\right)+2\left(40\right)\\ h\left(40\right)=-40+80\\ h\left(40\right)=40\end{array}$

Hence the coordinate of the vertex is $\left(40,40\right)$.

Step 3. Conclusion .

The equation of the axis of symmetry is $x=40$ and the coordinates of the vertex of the graph is $\left(40,40\right)$.