Write the quadratic function in standard form.
[tex]
\begin{array}{l}
f(x)=x^2-10 x+27 \\
f(x)=2 x^2+11 x+27
\end{array}
[/tex]
Sketch its graph.
Answer (1)
Convert the first quadratic function f ( x ) = x 2 − 10 x + 27 to standard form by completing the square: f ( x ) = ( x − 5 ) 2 + 2 .
Convert the second quadratic function f ( x ) = 2 x 2 + 11 x + 27 to standard form by completing the square: f ( x ) = 2 ( x + 4 11 ) 2 + 8 95 .
Identify the vertex of each parabola from the standard form: ( 5 , 2 ) and ( − 4 11 , 8 95 ) .
Sketch the graphs of the parabolas using their vertices and noting that both open upwards: f ( x ) = ( x − 5 ) 2 + 2 and f ( x ) = 2 ( x + 4 11 ) 2 + 8 95 .
Explanation
Understanding the Problem We are given two quadratic functions and asked to write them in standard form and sketch their graphs. The standard form of a quadratic function is given by f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola.
Completing the Square for the First Function For the first quadratic function, f ( x ) = x 2 − 10 x + 27 , we complete the square to write it in standard form. We have
f ( x ) = ( x 2 − 10 x ) + 27
To complete the square, we need to add and subtract ( 2 − 10 ) 2 = ( − 5 ) 2 = 25 inside the parenthesis:
f ( x ) = ( x 2 − 10 x + 25 − 25 ) + 27 f ( x ) = ( x 2 − 10 x + 25 ) − 25 + 27 f ( x ) = ( x − 5 ) 2 + 2
So the standard form of the first quadratic function is f ( x ) = ( x − 5 ) 2 + 2 . The vertex of this parabola is ( 5 , 2 ) .
Completing the Square for the Second Function For the second quadratic function, f ( x ) = 2 x 2 + 11 x + 27 , we complete the square to write it in standard form. We have
f ( x ) = 2 ( x 2 + 2 11 x ) + 27
To complete the square, we need to add and subtract ( 2 11/2 ) 2 = ( 4 11 ) 2 = 16 121 inside the parenthesis:
f ( x ) = 2 ( x 2 + 2 11 x + 16 121 − 16 121 ) + 27 f ( x ) = 2 ( x 2 + 2 11 x + 16 121 ) − 2 ( 16 121 ) + 27 f ( x ) = 2 ( x + 4 11 ) 2 − 8 121 + 27 f ( x ) = 2 ( x + 4 11 ) 2 + 8 216 − 121 f ( x ) = 2 ( x + 4 11 ) 2 + 8 95
So the standard form of the second quadratic function is f ( x ) = 2 ( x + 4 11 ) 2 + 8 95 . The vertex of this parabola is ( − 4 11 , 8 95 ) .
Sketching the Graphs Now we have the two quadratic functions in standard form:
f ( x ) = ( x − 5 ) 2 + 2 f ( x ) = 2 ( x + 4 11 ) 2 + 8 95
The first parabola has vertex ( 5 , 2 ) and opens upwards. The second parabola has vertex ( − 4 11 , 8 95 ) = ( − 2.75 , 11.875 ) and also opens upwards. To sketch the graphs, we can plot the vertices and find additional points by plugging in values for x .
Final Answer The quadratic functions in standard form are:
f ( x ) = ( x − 5 ) 2 + 2 f ( x ) = 2 ( x + 4 11 ) 2 + 8 95
Examples
Understanding quadratic functions and their standard form is crucial in various fields, such as physics and engineering. For example, when analyzing the trajectory of a projectile, the height of the projectile can be modeled by a quadratic function. By writing the quadratic function in standard form, we can easily determine the maximum height reached by the projectile and the time at which it occurs. This information is vital for making predictions and optimizing the projectile's motion.
