To graph the parabola we can give any numbers for x and then solve for y, or we can find the vertex and then go to either way.
0. Finding the vertex.
The x value of the vertex's coordinate can be calculated as follows:
![x=(-b)/(2a)](https://img.qammunity.org/2023/formulas/mathematics/high-school/thp2xvy4ibcymljghqfdd580j7p1ckgbwu.png)
where a and b represent the coefficients of the equation of the parabola in the form:
![ax^2+bx+c=0](https://img.qammunity.org/2023/formulas/mathematics/high-school/mvkhuzwnjhb4epaf7jjcoq2vi4zdi4350m.png)
As our parabola is:
![y=-x^2](https://img.qammunity.org/2023/formulas/mathematics/high-school/am49qw61yic44pgbeeukzfpgvu4vpdbnmq.png)
we can see that a = -1 and b = 0. Replacing this in the formula for the x coordinate:
![x=(0)/(2(-1))=0](https://img.qammunity.org/2023/formulas/mathematics/high-school/g11y258jahrjld7r71h92scdc43fp3aw6m.png)
We have to replace this value in the equation:
![y=-(0)^2=0](https://img.qammunity.org/2023/formulas/mathematics/high-school/70wtpb0tid2wa2fhhgce7th486p4gxk9zw.png)
Then the vertex can be found in (0,0).
Now we have to find different coordinates in each side to graph the parabola by giving arbitrarily numbers to x and evaluating them in the equation.
• x = 1
![y=-(1)^2=-1](https://img.qammunity.org/2023/formulas/mathematics/high-school/wee7wooq56e8i2xfslzmlz4ao0bc419ig8.png)
Then the coordinate, in this case, is (1, -1)
• x = -2
![y=-(-2)^2=-(4)=-4](https://img.qammunity.org/2023/formulas/mathematics/high-school/q1gph5cu4gxnracdr2w1ulgtmewbidihpy.png)
The coordinate, in this case, is (-2, -4).
• x = 3
![y=-(3)^2=-9](https://img.qammunity.org/2023/formulas/mathematics/college/34vwal4vjpy1vypbzyscyey2k9forx7ven.png)
The coordinate is (3, -9).
• x = -4
![y=-(-4)^2=-(16)=-16](https://img.qammunity.org/2023/formulas/mathematics/high-school/y6gk6f1y3a0ad5dl7xomgdajx9ynl2866x.png)
The coordinate is (-4, -16).
We can keep choosing any value for x to find y and get the coordinate in need be.
Answer:
• (0,0)
,
• (1, -1)
,
• (-2, -4)
,
• (3, -9)
,
• (-4,-16)