Question 1198783
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Answers for problem 1 in red
Vertex:  <font color=red>(-1, -7)</font>
Direction of opening: <font color=red>downward</font>
Axis of symmetry: <font color=red>x = -1</font>
min/max value: <font color=red>Max is y = -7</font>
range: <font color=red>y ≤  -7</font>
y-intercept:  <font color=red>-7.5</font>



Explanation:


Compare y = -0.5(x+1)^2-7 to y = a(x-h)^2+k
We find that h = -1 and k = -7
Think of x+1 as x-(-1)
Therefore, the vertex is (h,k) = <font color=red>(-1, -7)</font>


The value of 'a' determines directly how the parabola opens. 
If a < 0, then the parabola opens downward.
If a > 0, then the parabola opens upward.
In this case we have a = -0.5 to indicate the parabola opens <font color=red>downward</font>.


The axis of symmetry is of the form x = h, where h was the x coordinate of the vertex. 
The axis of symmetry is the vertical mirror line through the vertex.
Therefore, h = -1 leads to <font color=red>x = -1</font> being the axis of symmetry.


Recall in the second paragraph we discussed a = -0.5 indicating the parabola opens downward. 
This causes the vertex to be the highest point, aka <font color=red>max</font>. 
The max value is the largest possible y output which in this case is <font color=red>y = -7</font>. 
This is the highest the parabola can go before falling back down.


Refer to the previous paragraph above. We found that y = -7 is the largest output possible. 
The range is the set of possible y outputs. Either y = -7 or y < -7.
In short, the range is the set of y values such that {{{y <= -7}}}


To find the y-intercept, plug in x = 0 and compute.
y = -0.5(x+1)^2 - 7
y = -0.5(0+1)^2 - 7
y = -7.5
The y-intercept is <font color=red>-7.5</font>; placing it at the location (0,-7.5)


Graph:
{{{drawing(500,500,-8,10,-15,4,
graph(500,500,-8,10,-15,4,-100,-100,-0.5(x+1)^2-7),
circle(-1,-7,-0.1),
circle(-1,-7,-0.12),
circle(-1,-7,-0.15),
line(-4,-6+0.2,-1.5,-6.8),
line(-1.5,-6.8,-1.5-0.3,-6.8+0.5),
line(-1.5,-6.8,-1.5-0.5,-6.8-0.1),
locate(-6,-5,"Vertex = (-1,-7)"),
circle(0,-7.5,-0.1),
circle(0,-7.5,-0.12),
circle(0,-7.5,-0.15),
locate(0.6,-7,matrix(1,2,"y-intercept","(0,-7.5)"))
)}}}
I recommend using either Desmos or GeoGebra as a graphing tool.
Both of which are free.



I'll leave problem 2 for you to complete. 
If you're still stuck, then let me know or make a new post on this website.
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