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What is the y-coordinate of a vertex of a parabola with the following equation $ y = {x^2} - 8x + 18 $ ?

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Last updated date: 24th Jul 2024
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Answer
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Hint: To simplify this question , we need to solve it step by step . In order to solve and write the find the vertex of a quadratic equation . We will use the vertex formula and determine the y- coordinate of the vertex of a parabola in simplest form by using the formula $ \left( { - \dfrac{b}{{2a}},f(x)} \right) $ . We will first determine the x-coordinate and then we fill substitute the found x-coordinate in the quadratic equation to determine y-coordinate of the vertex of a parabola with the following equation $ y = {x^2} - 8x + 18 $
We can also use the quadratic in vertex form method to get our required result.

Complete step-by-step answer:
A quadratic equation is written as $ a{x^2} + bx + c $ in its standard form . As we know that in the quadratic equation , a is the coefficient of the first term in the quadratic, b is the coefficient of the second term and c is the coefficient of the third term in the quadratic.
Now we are Assessing the coefficients of our quadratic equation given $ y = {x^2} - 8x + 18 $
The values of a, b and c are as follows =.
 $
  a = 1 \\
  b = - 8 \\
  c = 18 \;
  $
And the vertex can be found by using the formula $ \left( { - \dfrac{b}{{2a}}} \right) $
Vertex = $ \left( { - \dfrac{b}{{2a}},f(x)} \right) $
Now , we will substitute the values in this formula we get
  $
  \left( { - \dfrac{b}{{2a}}} \right) \\
   \Rightarrow - \dfrac{{( - 8)}}{{2 \times 1}} \\
   \Rightarrow \dfrac{8}{2} \\
   \Rightarrow 4 \;
  $
This 4 is the required x-coordinate of the vertex of parabola .
Now , looking at the formula of Vertex $ \left( { - \dfrac{b}{{2a}},f(x)} \right) $ , we got x- coordinate . We will be substituting the x-coordinate into the given quadratic equation $ y = {x^2} - 8x + 18 $ to get the required y-coordinate of the vertex of the parabola

By Substituting , we get-
Vertex = $ \left( { - \dfrac{b}{{2a}},f(x)} \right) $
This formula becomes and when simplifying further , we will get vertex .
 
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Vertex = $ (4,2) $
Therefore , $ (4,2) $ is the required vertex and 2 is the required y-coordinate of the vertex of the parabola with the quadratic equation $ y = {x^2} - 8x + 18 $ .

Note: If \[a > 0\], the parabola opens upward. If \[a < 0\] , the parabola opens downward.
The general form of a parabola to find the equation for the axis of symmetry.
The axis of symmetry is defined by $ x = \left( { - \dfrac{b}{{2a}}} \right) $ .
The vertex of a parabola will be either a maximum or a minimum, the range will consist of all y-values greater than or equal to the y-coordinate of the vertex or less than or equal to the y-coordinate at the turning point, depending on whether the parabola opens up or down.
Always try to understand the mathematical statement carefully and keep things distinct .