The coordinates of the vertex is \(\boxed{\bf (-5,-28)}\).
Further explanation:
Given:
The function \(f(x)\) is as follows:
\(\boxed{f(x)=x^{2}+10x-3}\)
Vertex form of the function \(f(x)\) is as follows:
\(\boxed{f(x)=(x+5)^{2}-28}\)
Definition of vertex form:
The vertex form of a quadratic function is as follows:
\(\boxed{f(x)=a(x-h)^{2}+k}\) …… (1)
Here, \((h,k)\) is the vertex of the parabola.
The standard form of the quadratic equation is as follows:
\(\boxed{f(x)=ax^{2}+bx+c}\)
If the function \(f(x)\) is written in vertex form then \((h,k)\) is the vertex of the parabola and \(x=h\) is the axis of symmetry.
Calculation:
To convert the function \(f(x)\) from standard form to vertex form first convert the function \(f(x)\) into the complete square form as shown below,
\(\begin{aligned}f(x)&=x^{2}+10x-3\\&=x^{2}+(2\cdot x\cdot 5)-3+5^{2}-5^{2}\\&=x^{2}+(2\cdot x\cdot 5)+5^{2}-3-25\\&=(x+5)^{2}-28\end{aligned}\)
Label the above equation as follows:
\(\boxed{f(x)=(x+(-5))^{2}-28}\) ......(2)
Compare equation (1) and equation (2) to obtain the value of \(h\) and \(k\).
The value of \(h\) and \(k\) are as follows:
\(\boxed{\begin{aligned}h&=-5\\k&=-28\end{aligned}}\)
Therefore, the vertex of the parabola is at the point \((-5,-28)\).
A direct formula to find the vertex \((h,k)\) of the parabola is,
\(\boxed{h=-\dfrac{b}{2a}\text{ and }k=f(h)}\)
The horizontal shift \(h\) is calculated as follows:
\(\begin{aligned}h&=-\dfrac{10}{2\cdot 1}\\&=\dfrac{-10}{2}\\&=-5\end{aligned}\)
The vertical shift \(k\) is calculated as follows:
\(\begin{aligned}k&=f(-5)\\&=(-5)^{2}+(10\cdot (-5)-3)\\&=25-50-3\\&=-25-3\\&=-28\end{aligned}\)
Therefore, the coordinates of the vertex is at the point \((-5,-28)\).
Thus, the coordinates of the vertex is \(\boxed{\bf (-5,-28)}\).
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.