\[ \boldsymbol{ f ( x ) \ = \ 2 x^{ 2 } \ – \ 8 x \ + \ 3 } \]
The aim of this question is to learn how to evaluate the vertex location of a parabola.
A U-shaped curve that follows the quadratic law (its equation is quadratic), is called a parabola. A parabola has a mirror like symmetry. The point on a parabolic curve that touches its symmetrical axis is called a vertex. Given a parabola of the form:
\[ f ( x ) \ = \ a x^{ 2 } \ + \ b x \ + \ c \]
The x-coordinate of its vertex can be evaluated by using the following formula:
\[ h \ = \ \dfrac{ – b }{ 2a } \]
Expert Answer
Given that:
\[ f ( x ) \ = \ 2 x^{ 2 } \ – \ 8 x \ + \ 3 \]
Comparing with the standard form of quadratic equation, we can conclude that:
\[ a \ = \ 2 \]
\[ b \ = \ -8 \]
\[ c \ = \ 3 \]
Recall the standard formula for the x-coordinate of the vertex of a parabola:
\[ h \ = \ \dfrac{ – b }{ 2a } \]
Substituting values:
\[ h \ = \ \dfrac{ – ( -8 ) }{ 2 ( 2 ) } \]
\[ \Rightarrow h \ = \ \dfrac{ 8 }{ 4 } \]
\[ \Rightarrow h \ = \ 2 \]
To find the y-coordinate, we simply evaluate the given equation of the parabola at x = 2. Recall:
\[ f ( x ) \ = \ 2 x^{ 2 } \ – \ 8 x \ + \ 3 \]
Substituting x = 2 in the above equation:
\[ f ( 2 ) \ = \ 2 ( 2 )^{ 2 } \ – \ 8 ( 2 ) \ + \ 3 \]
\[ \Rightarrow f ( 2 ) \ = \ 2 ( 4 ) \ – \ 8 ( 2 ) \ + \ 3 \]
\[ \Rightarrow f( 2 ) \ = \ 8 \ – \ 16 \ + \ 3 \]
\[ \Rightarrow f ( 2 ) \ = \ -5 \]
Hence, the vertex is located at (2, -5).
Numerical Result
The vertex is located at (2, -5).
Example
Given the following equation of a parabola, find the location of its vertex.
\[ \boldsymbol{ f ( x ) \ = \ x^{ 2 } \ – \ 2 x \ + \ 1 } \]
For the x-coordinate of the vertex:
\[ h \ = \ \dfrac{ – ( -2 ) }{ 2 ( 1 ) } \]
\[ \Rightarrow h \ = \ \dfrac{ 2 }{ 2 } \]
\[ \Rightarrow h \ = \ 1 \]
To find the y-coordinate, we simply evaluate the given equation of the parabola at x = 1. Recall:
\[ f ( 2 ) \ = \ ( 1 )^{ 2 } \ – \ 2 ( 1 ) \ + \ 1 \]
\[ \Rightarrow f( 2 ) \ = \ 1 \ – \ 2 \ + \ 1 \]
\[ \Rightarrow f ( 2 ) \ = \ 0 \]
Hence, the vertex is located at (1, 0).