Question

11. Find a formula for the quadratic function whose graph is

shown.

Short answer

The required quadratic equation is \(f\left( x \right) = 2{x^2} - 12x + 18\).

Step-by-step solution

Observe the given graph

It is known that the quadratic function is of the form \(f\left( x \right) = a{x^2} + bx + c\).

The given graph has the\(y\)intercept of 18, so\(f\left( 0 \right) = 18\).

Also, the points\(\left( {3,0} \right)\)and\(\left( {4,2} \right)\)exist on the curve of the given graph. So, these points must satisfy the quadratic function.

Obtain the intercept

Use \(f\left( 0 \right) = 18\) to find the value of \(c\) as shown below:

\(\begin{aligned}f\left( 0 \right) &= 18\\a{\left( 0 \right)^2} + b\left( 0 \right) + c &= 18\\c &= 18\end{aligned}\)

Thus, \(c = 18\).

Obtain the equations

Plug-in points \(\left( {3,0} \right)\)and \(\left( {4,2} \right)\) in the function \(f\left( x \right)\) as shown below:

As the points\(\left( {3,0} \right)\)and\(\left( {4,2} \right)\)satisfy\(f\left( x \right) = a{x^2} + bx + c\). So, take\(f\left( 3 \right) = 0\)and \(f\left( 4 \right) = 2\):

\(\begin{aligned}f\left( 3 \right) &= 0\\a{\left( 3 \right)^2} + b\left( 3 \right) + 18 &= 0\\9a + 3b &= - 18\\3a + b &= - 6\end{aligned}\)

And,

\(\begin{aligned}f\left( 4 \right) &= 2\\a{\left( 4 \right)^2} + b\left( 4 \right) + 18 &= 2\\16a + 4b &= - 16\\4a + b &= - 4\end{aligned}\)

Solve the equations for \(a,\,{\rm{ }}b\)

Subtract the equations \(3a + b = - 6\), and \(4a + b = - 4\) to get the value of \(a\).

\(\begin{aligned}3a + b - \left( {4a + b} \right) &= - 6 - \left( { - 4} \right)\\ - a &= - 2\\a &= 2\end{aligned}\)

Put \(a = 2\) in the equation \(4a + b = - 4\), we can calculate \(b\)as shown below:

\(\begin{aligned}4\left( 2 \right) + b &= - 4\\8 + b &= - 4\\b &= - 12\end{aligned}\)

So, \(a = 2\) and \(b = - 12\).

Get the quadratic equation

On plugging the values of \(a,b,c\) into \(f\left( x \right) = a{x^2} + bx + c\), the required quadratic equation is \(f\left( x \right) = 2{x^2} - 12x + 18\).

Thus, the quadratic equation is \(f\left( x \right) = 2{x^2} - 12x + 18\).