Question

on this slide you will need to fill out the table and graph out a parabola on the coordinate plane provided. any work you need to do can be done below in the space provided.
graph the quadratic equation and identify the solution(s).

1. $y = x^2 - 6x + 9$

$x$ $y$
solutions:

1. $y = x^2 + 4x + 9$

$x$ $y$
solutions:

1. $y = 2x^2 - 4x + 3$

$x$ $y$
solutions:

1. $y = \frac{1}{2}x^2 - 4x + 9$

$x$ $y$
solutions:

1. $y = 2x^2 - 12x + 19$

$x$ $y$
solutions:

1. $y = -2x^2 - 8x - 10$

$x$ $y$
solutions:
do your work here. provide the number for the problem above so i can identify which work corresponds to which problem:

Explanation:

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Problem 8: $y = x^2 - 6x + 9$

Step1: Find vertex x-value

$x = -\frac{b}{2a} = -\frac{-6}{2(1)} = 3$

Step2: Calculate y-values

For $x=1$: $y=(1)^2-6(1)+9=4$
For $x=2$: $y=(2)^2-6(2)+9=1$
For $x=3$: $y=(3)^2-6(3)+9=0$
For $x=4$: $y=(4)^2-6(4)+9=1$
For $x=5$: $y=(5)^2-6(5)+9=4$

Step3: Find solution (x-intercept)

Set $y=0$: $x^2-6x+9=(x-3)^2=0 \implies x=3$

Table:

$x$	$y$
2	1
3	0
4	1
5	4

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Problem 9: $y = x^2 + 4x + 9$

Step1: Find vertex x-value

$x = -\frac{b}{2a} = -\frac{4}{2(1)} = -2$

Step2: Calculate y-values

For $x=-4$: $y=(-4)^2+4(-4)+9=9$
For $x=-3$: $y=(-3)^2+4(-3)+9=6$
For $x=-2$: $y=(-2)^2+4(-2)+9=5$
For $x=-1$: $y=(-1)^2+4(-1)+9=6$
For $x=0$: $y=(0)^2+4(0)+9=9$

Step3: Find solution (x-intercept)

Discriminant: $b^2-4ac=16-36=-20<0$, no real solutions

Table:

$x$	$y$
-3	6
-2	5
-1	6
0	9

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Problem 10: $y = 2x^2 - 4x + 3$

Step1: Find vertex x-value

$x = -\frac{b}{2a} = -\frac{-4}{2(2)} = 1$

Step2: Calculate y-values

For $x=-1$: $y=2(-1)^2-4(-1)+3=9$
For $x=0$: $y=2(0)^2-4(0)+3=3$
For $x=1$: $y=2(1)^2-4(1)+3=1$
For $x=2$: $y=2(2)^2-4(2)+3=3$
For $x=3$: $y=2(3)^2-4(3)+3=9$

Step3: Find solution (x-intercept)

Discriminant: $b^2-4ac=16-24=-8<0$, no real solutions

Table:

$x$	$y$
0	3
1	1
2	3
3	9

---

Problem 11: $y = \frac{1}{2}x^2 - 4x + 9$

Step1: Find vertex x-value

$x = -\frac{b}{2a} = -\frac{-4}{2(\frac{1}{2})} = 4$

Step2: Calculate y-values

For $x=2$: $y=\frac{1}{2}(2)^2-4(2)+9=3$
For $x=3$: $y=\frac{1}{2}(3)^2-4(3)+9=1.5$
For $x=4$: $y=\frac{1}{2}(4)^2-4(4)+9=1$
For $x=5$: $y=\frac{1}{2}(5)^2-4(5)+9=1.5$
For $x=6$: $y=\frac{1}{2}(6)^2-4(6)+9=3$

Step3: Find solution (x-intercept)

Discriminant: $b^2-4ac=16-18=-2<0$, no real solutions

Table:

$x$	$y$
3	1.5
4	1
5	1.5
6	3

---

Problem 13: $y = 2x^2 - 12x + 19$

Step1: Find vertex x-value

$x = -\frac{b}{2a} = -\frac{-12}{2(2)} = 3$

Step2: Calculate y-values

For $x=1$: $y=2(1)^2-12(1)+19=9$
For $x=2$: $y=2(2)^2-12(2)+19=3$
For $x=3$: $y=2(3)^2-12(3)+19=1$
For $x=4$: $y=2(4)^2-12(4)+19=3$
For $x=5$: $y=2(5)^2-12(5)+19=9$

Step3: Find solution (x-intercept)

Discriminant: $b^2-4ac=144-152=-8<0$, no real solutions

Table:

$x$	$y$
2	3
3	1
4	3
5	9

---

Problem 14: $y = -2x^2 - 8x - 10$

Step1: Find vertex x-value

$x = -\frac{b}{2a} = -\frac{-8}{2(-2)} = -2$

Step2: Calculate y-values

For $x=-4$: $y=-2(-4)^2-8(-4)-10=-10$
For $x=-3$: $y=-2(-3)^2-8(-3)-10=-4$
For $x=-2$: $y=-2(-2)^2-8(-2)-10=-2$
For $x=-1$: $y=-2(-1)^2-8(-1)-10=-4$
For $x=0$: $y=-2(0)^2-8(0)-10=-10$

Step3: Find solution (x-intercept)

Discriminant: $b^2-4ac=64-80=-16<0$, no real solutions

Table:

$x$	$y$
-3	-4
-2	-2
-1	-4
0	-10

---

Answer:

1. Problem 8

Table:

$x$	$y$
2	1
3	0
4	1
5	4

Solution: $x=3$ (repeated root)
Graph: Plot the points; parabola opens up, touches the x-axis at $(3,0)$

1. Problem 9

Table:

$x$	$y$
-3	6
-2	5
-1	6
0	9

Solution: No real solutions
Graph: Plot the points; parabola opens up, lies entirely above the x-axis

1. Problem 10

Table:

$x$	$y$
0	3
1	1
2	3
3	9

Solution: No real solutions
Graph: Plot the points; parabola opens up, lies entirely above the x-axis

1. Problem 11

Table:

$x$	$y$
3	1.5
4	1
5	1.5
6	3

Solution: No real solutions
Graph: Plot the points; parabola opens up, lies entirely above the x-axis

1. Problem 13

Table:

$x$	$y$
2	3
3	1
4	3
5	9

Solution: No real solutions
Graph: Plot the points; parabola opens up, lies entirely above the x-axis

1. Problem 14

Table:

$x$	$y$
-3	-4
-2	-2
-1	-4
0	-10

Solution: No real solutions
Graph: Plot the points; parabola opens down, lies entirely below the x-axis