Question

Find expressions for the quadratic functions whose graphs

are shown

Short answer

The expression of the quadratic function for the first graph is\(f\left( x \right) = {\left( {x - 3} \right)^2}\)

The expression of the quadratic function for the second graph is\(g\left( x \right) = \left( { - {x^2} - 2.5x + 1} \right)\)

Step-by-step solution

Quadratic function

A quadratic function is a function that has one or more variables, but in which the highest exponent of the variable is two.

Finding the expression of quadratic function for the first graph

Let the quadratic function is,

\(f\left( x \right) = a{\left( {x - b} \right)^2}\)

From the graph of the function, we know

\(x = 3,\,\,f\left( 3 \right) = 0\)

Substituting the values into the equation

\(\begin{array}{c}f\left( 3 \right) = a{\left( {3 - b} \right)^2}\\0 = a{\left( {3 - b} \right)^2}\\b = 3\end{array}\)

Now again we know

\(x = 4,\,\,f\left( 2 \right) = 0\)

\(\begin{array}{c}f\left( 4 \right) = a{\left( {4 - b} \right)^2}\\2 = a{\left( {4 - b} \right)^2}\end{array}\)

Now substituting the value,\(b = 3\)

\(\begin{array}{c}2 = a{\left( {4 - 3} \right)^2}\\a = 1\end{array}\)

Substituting the values ofa andb in the function.

The expression of the quadratic function is \(f\left( x \right) = {\left( {x - 3} \right)^2}\).

Finding the expression of quadratic function for the second graph

Let quadratic function is

\(g\left( x \right) = a{x^2} + bx + c\)

From the graph of the function, we know

\(x = 0,\,\,g\left( 0 \right) = 1\)

Therefore,

\(\begin{array}{c}g\left( 0 \right) = a{\left( 0 \right)^2} + b\left( 0 \right) + c\\1 = c\end{array}\)

Substituting this value into quadratic function

\(g\left( x \right) = a{x^2} + bx + 1\)

From graph we know

\(x = - 2,\,\,g\left( { - 2} \right) = 2\)

Substituting the values

\(\begin{array}{c}g\left( { - 2} \right) = a{\left( { - 2} \right)^2} + b\left( { - 2} \right) + 1\\2 = 4a - 2b + 1\\4a - 2b = 1\,\,\,\,\,\,\,\,......\left( 1 \right)\end{array}\)

From graph we know

\(x = 1,\,\,g\left( 1 \right) = - 2.5\)

Substituting the values

\(\begin{array}{c}g\left( 1 \right) = a{\left( 1 \right)^2} + b\left( 1 \right) + 1\\ - 2.5 = a + b + 1\\a + b = - 3.5\\a = - 3.5 - b\end{array}\)

Substituting the value ofa into the equation (i) and solving forb

\(\begin{array}{c}4\left( { - 3.5 - b} \right) - 2b = 1\\ - 14 - 4b - 2b = 1\\b = - 2.5\end{array}\)

Then

\(\begin{array}{l}a = - 3.5 - \left( { - 2.5} \right)\\a = - 1\end{array}\)

Substituting the values ofa andb in the function

Hence the expression of the quadratic function is \(g\left( x \right) = - {x^2} - 2.5x + 1\)