Respuesta :
Answer:
The y-intercept is y = -4.
The x-intercepts are [tex]x = 4[/tex] and [tex]x = -\frac{1}{2}[/tex]
The vertex is [tex](\frac{7}{4},-\frac{81}{8})[/tex]
Step-by-step explanation:
Quadratic equation:
Has the following format:
[tex]y = ax^2 + bx + c[/tex]
The y-intercept is c.
Finding the x-intercepts:
Given a second order polynomial expressed by the following equation:
[tex]ax^{2} + bx + c, a\neq0[/tex].
This polynomial has roots [tex]x_{1}, x_{2}[/tex] such that [tex]ax^{2} + bx + c = a(x - x_{1})*(x - x_{2})[/tex], given by the following formulas:
[tex]x_{1} = \frac{-b + \sqrt{\Delta}}{2*a}[/tex]
[tex]x_{2} = \frac{-b - \sqrt{\Delta}}{2*a}[/tex]
[tex]\Delta = b^{2} - 4ac[/tex]
Vertex:
Suppose we have a quadratic function in the following format:
[tex]f(x) = ax^{2} + bx + c[/tex]
It's vertex is the point [tex](x_{v}, y_{v})[/tex]
In which
[tex]x_{v} = -\frac{b}{2a}[/tex]
[tex]y_{v} = -\frac{\Delta}{4a}[/tex]
Where
[tex]\Delta = b^2-4ac[/tex]
In this question:
The quadratic equation is [tex]2x^2 - 7x - 4[/tex], which has [tex]a = 2, b = -7, c = -4[/tex]. This means that the y-intercept is y = -4.
x-intercepts:
[tex]\Delta = (-7)^2-4(2)(-4) = 81[/tex]
[tex]x_{1} = \frac{-(-7) + \sqrt{81}}{2*2} = 4[/tex]
[tex]x_{2} = \frac{-(-7) - \sqrt{81}}{2*2} = -\frac{1}{2}[/tex]
The x-intercepts are [tex]x = 4[/tex] and [tex]x = -\frac{1}{2}[/tex]
Vertex:
[tex]x_{v} = -\frac{(-7)}{2*2} = \frac{7}{4}[/tex]
[tex]y_{v} = -\frac{81}{4(2)} = -\frac{81}{8}[/tex]
The vertex is [tex](\frac{7}{4},-\frac{81}{8})[/tex]
