Equation of the Parabola.
If the vertex of a parabola is located at the point (h, k), then the equation of the parabola is written as:
[tex]y=a(x-h)^2+k[/tex]
Where a is the leading coefficient. To find the coordinates of the vertex, given its equation, we use the 'square completion' technique.
We are given the equation:
[tex]y=x^2-4x-21[/tex]
We need to transform this equation so it can be expressed like the general equation given above. Adding and subtracting 4:
[tex]y=x^2-4x+4-21-4[/tex]
Rearranging:
[tex]y=(x^2-4x+4)-25[/tex]
Factoring:
[tex]y=(x-2)^2-25[/tex]
The vertex of the parabola is located at (2, -25).
To find the x-intercepts, we set y=0 and solve the equation:
[tex]\begin{gathered} (x-2)^2-25=0 \\ \text{Add 25:} \\ (x-2)^2=25 \end{gathered}[/tex]
Taking the square root (it has two signs):
[tex]x-2=\pm5[/tex]
Solving for x, we get two possible answers:
x = 5 + 2 = 7
x = -5 + 2 = -3
X-inercepts: -3, 7
