Respuesta :
Answer:
Option A is correct
(–5, –28)
Step-by-step explanation:
For a quadratic equation:
[tex]y=ax^2+bx+c[/tex]
We know that convert of quadratic into vertex form, using the method of completing the square is,
[tex]y = a(x-h)^2+k[/tex]
where,
(h, k) is the vertex.
As per the statement:
The function [tex]f(x) = x^2 + 10x -3[/tex]
written in vertex form is:
[tex]f(x) = (x+5)^2-28[/tex]
From the above definition we have;
h = -5 and k = -28
Therefore, the coordinates of the vertex is, (-5, -28)
The coordinates of the vertex is [tex]\boxed{\bf (-5,-28)}[/tex].
Further explanation:
Given:
The function [tex]f(x)[/tex] is as follows:
[tex]\boxed{f(x)=x^{2}+10x-3}[/tex]
Vertex form of the function [tex]f(x)[/tex] is as follows:
[tex]\boxed{f(x)=(x+5)^{2}-28}[/tex]
Definition of vertex form:
The vertex form of a quadratic function is as follows:
[tex]\boxed{f(x)=a(x-h)^{2}+k}[/tex] …… (1)
Here, [tex](h,k)[/tex] is the vertex of the parabola.
The standard form of the quadratic equation is as follows:
[tex]\boxed{f(x)=ax^{2}+bx+c}[/tex]
If the function [tex]f(x)[/tex] is written in vertex form then [tex](h,k)[/tex] is the vertex of the parabola and [tex]x=h[/tex] is the axis of symmetry.
Calculation:
To convert the function [tex]f(x)[/tex] from standard form to vertex form first convert the function [tex]f(x)[/tex] into the complete square form as shown below,
[tex]\begin{aligned}f(x)&=x^{2}+10x-3\\&=x^{2}+(2\cdot x\cdot 5)-3+5^{2}-5^{2}\\&=x^{2}+(2\cdot x\cdot 5)+5^{2}-3-25\\&=(x+5)^{2}-28\end{aligned}[/tex]
Label the above equation as follows:
[tex]\boxed{f(x)=(x+(-5))^{2}-28}[/tex] ......(2)
Compare equation (1) and equation (2) to obtain the value of [tex]h[/tex] and [tex]k[/tex].
The value of [tex]h[/tex] and [tex]k[/tex] are as follows:
[tex]\boxed{\begin{aligned}h&=-5\\k&=-28\end{aligned}}[/tex]
Therefore, the vertex of the parabola is at the point [tex](-5,-28)[/tex].
A direct formula to find the vertex [tex](h,k)[/tex] of the parabola is,
[tex]\boxed{h=-\dfrac{b}{2a}\text{ and }k=f(h)}[/tex]
The horizontal shift [tex]h[/tex] is calculated as follows:
[tex]\begin{aligned}h&=-\dfrac{10}{2\cdot 1}\\&=\dfrac{-10}{2}\\&=-5\end{aligned}[/tex]
The vertical shift [tex]k[/tex] is calculated as follows:
[tex]\begin{aligned}k&=f(-5)\\&=(-5)^{2}+(10\cdot (-5)-3)\\&=25-50-3\\&=-25-3\\&=-28\end{aligned}[/tex]
Therefore, the coordinates of the vertex is at the point [tex](-5,-28)[/tex].
Thus, the coordinates of the vertex is [tex]\boxed{\bf (-5,-28)}[/tex].
Learn more:
1. Learn more about vertex form https://brainly.com/question/1286775
2. Learn more about quadratic function https://brainly.com/question/1332667
Answer details:
Grade: High school
Subject: Mathematics
Chapter: Quadratic function
Keywords: Function, vertex form, quadratic function, coordinates, x2+10x-3, f(x) standard form, parabola, vertical shift, horizontal shift, degree 2, highest power is 2, quadratic equation.
