Question

practice
for each equation, state which way the parabola opens and how much the vertex shifts from the origin. make a table of values. then graph each equation.

1. $y=x^{2}$
2. $y=3x^{2}+1$
3. $y=-4x^{2}$
4. $y=\frac{1}{2}x^{2}$
5. $y=-\frac{1}{2}x^{2}-3$
6. $y=x^{2}+\frac{1}{2}$
7. $y=2x^{2}-4$
8. $y=-x^{2}-3$
9. $y=-4x^{2}+7$
10. $y=\frac{1}{4}x^{2}-2$
11. $y=9x^{2}$
12. $y+3=0.25x^{2}$

Explanation:

Step1: Analyze parabola direction/shift

For a quadratic $y=ax^2+k$:

• If $a>0$, opens up; $a<0$, opens down.
• Vertical shift: $k$ units up if $k>0$, down if $k<0$.

---

1. $y=x^2$

Step1: Identify $a,k$

$a=1>0$, $k=0$

Step2: Table of values

$x$	$y=x^2$
-1	1
0	0
1	1
2	4

2. $y=3x^2+1$

Step1: Identify $a,k$

$a=3>0$, $k=1$

Step2: Table of values

$x$	$y=3x^2+1$
-1	4
0	1
1	4
2	13

3. $y=-4x^2$

Step1: Identify $a,k$

$a=-4<0$, $k=0$

Step2: Table of values

$x$	$y=-4x^2$
-1	-4
0	0
1	-4
2	-16

4. $y=\frac{1}{2}x^2$

Step1: Identify $a,k$

$a=\frac{1}{2}>0$, $k=0$

Step2: Table of values

$x$	$y=\frac{1}{2}x^2$
-1	$\frac{1}{2}$
0	0
1	$\frac{1}{2}$
2	2

5. $y=-\frac{1}{2}x^2-3$

Step1: Identify $a,k$

$a=-\frac{1}{2}<0$, $k=-3$

Step2: Table of values

$x$	$y=-\frac{1}{2}x^2-3$
-1	$-\frac{7}{2}$
0	-3
1	$-\frac{7}{2}$
2	-5

6. $y=x^2+\frac{1}{2}$

Step1: Identify $a,k$

$a=1>0$, $k=\frac{1}{2}$

Step2: Table of values

$x$	$y=x^2+\frac{1}{2}$
-1	$\frac{3}{2}$
0	$\frac{1}{2}$
1	$\frac{3}{2}$
2	$\frac{9}{2}$

7. $y=2x^2-4$

Step1: Identify $a,k$

$a=2>0$, $k=-4$

Step2: Table of values

$x$	$y=2x^2-4$
-1	-2
0	-4
1	-2
2	4

8. $y=-x^2-3$

Step1: Identify $a,k$

$a=-1<0$, $k=-3$

Step2: Table of values

$x$	$y=-x^2-3$
-1	-4
0	-3
1	-4
2	-7

9. $y=-4x^2+7$

Step1: Identify $a,k$

$a=-4<0$, $k=7$

Step2: Table of values

$x$	$y=-4x^2+7$
-1	3
0	7
1	3
2	-9

10. $y=\frac{1}{4}x^2-2$

Step1: Identify $a,k$

$a=\frac{1}{4}>0$, $k=-2$

Step2: Table of values

$x$	$y=\frac{1}{4}x^2-2$
-2	-1
0	-2
2	-1
4	2

11. $y=9x^2$

Step1: Identify $a,k$

$a=9>0$, $k=0$

Step2: Table of values

$x$	$y=9x^2$
-1	9
0	0
1	9
2	36

12. $y+3=0.25x^2$ (rewrite to $y=0.25x^2-3$)

Step1: Identify $a,k$

$a=0.25>0$, $k=-3$

Step2: Table of values

$x$	$y=0.25x^2-3$
-2	-2
0	-3
2	-2
4	1

Answer:

1. Opens up, no vertex shift. Table and graph as above.
2. Opens up, vertex shifts 1 unit up. Table and graph as above.
3. Opens down, no vertex shift. Table and graph as above.
4. Opens up, no vertex shift. Table and graph as above.
5. Opens down, vertex shifts 3 units down. Table and graph as above.
6. Opens up, vertex shifts $\frac{1}{2}$ unit up. Table and graph as above.
7. Opens up, vertex shifts 4 units down. Table and graph as above.
8. Opens down, vertex shifts 3 units down. Table and graph as above.
9. Opens down, vertex shifts 7 units up. Table and graph as above.
10. Opens up, vertex shifts 2 units down. Table and graph as above.
11. Opens up, no vertex shift. Table and graph as above.
12. Opens up, vertex shifts 3 units down. Table and graph as above.