Answer:
Vertex = (-2, - 9)
X- intercept: (-5, 0), (1, 0)
y-intercept: (0, -5)
Explanation:
If we have a quadratic equation of the form
[tex]y=(x-k)^2+h[/tex]then the vertex is given by
[tex]\text{vertex}=(k,h)[/tex]Now, in our case we have
[tex]y=(x+2)^2-9[/tex]meaning k = -2 and h = - 9; therefore, the vertex is at
[tex]vertex=(-2,9)[/tex]The intercepts of the parabola are the points where it intersects the x-axis. This happens when y = 0.
Putting in y = 0 in the equation for the parabola gives
[tex]0=(x+2)^2-9[/tex]adding 9 to both sides gives
[tex](x+2)^2=9[/tex]taking the square root of both sides gives
[tex]\sqrt[]{(x+2)^2}=\sqrt[]{9}[/tex][tex]x+2=\pm3[/tex]subtracting 2 from both sides gives
[tex]x=\pm3-2[/tex]which gives us two solutions
[tex]\begin{gathered} x=-3-2=-5 \\ x=3-2=1 \end{gathered}[/tex]Hence, the x-intercepts of the parabola are at
[tex]\begin{gathered} (-5,0) \\ (1,0) \end{gathered}[/tex]Now, we find the y-intercept.
The y-intercept is the point at which the parabola intersects the y-axis.
This happens when x = 0.
Putting in x = 0 in the equation for the parabola gives
[tex]\begin{gathered} y=(x+2)^2-9 \\ y=(0+2)^2-9 \\ y=4-9 \\ \boxed{y=-5} \end{gathered}[/tex]Hence, the y-intercept is at
[tex](0,-5)[/tex]To summarize our results,
Vertex = (-2, - 9)
X- intercept: (-5, 0), (1, 0)
y-intercept: (0, -5)
