Question

using tools in exercises 35–40, match the function with its graph. explain your reasoning.

1. ( g(x) = 2(x - 1)^2 - 2 ) 36. ( g(x) = \frac{1}{2}(x + 1)^2 - 2 )
2. ( g(x) = -2(x - 1)^2 + 2 )
3. ( g(x) = 2(x + 1)^2 + 2 ) 39. ( g(x) = -2(x + 1)^2 - 2 )
4. ( g(x) = 2(x - 1)^2 + 2 )

a.
graph a: a parabola opening upwards with vertex near (0, 2) on a coordinate grid from -4 to 4 on x and -4 to 4 on y
b.
graph b: a parabola opening upwards with vertex near (0, -2) on a coordinate grid from -4 to 4 on x and -4 to 4 on y
c.
graph c: a parabola opening upwards with vertex near (0, -2) on a coordinate grid from -4 to 4 on x and -4 to 4 on y, different from b
d.
graph d: a parabola opening downwards with vertex near (0, 2) on a coordinate grid from -4 to 4 on x and -4 to 4 on y
e.
graph e: a parabola opening upwards with vertex near (-2, 2) on a coordinate grid from -4 to 4 on x and -4 to 4 on y
f.
graph f: a parabola opening downwards with vertex near (-2, -2) on a coordinate grid from -4 to 4 on x and -4 to 4 on y

Explanation:

We use the vertex form of a quadratic function: $g(x) = a(x-h)^2 + k$, where:

• $(h,k)$ is the vertex
• $a>0$ opens upward, $a<0$ opens downward
• $|a|>1$ narrows the parabola, $0<|a|<1$ widens it

---

Problem 35: $g(x)=2(x-1)^2-2$

Step1: Identify vertex and direction

Vertex $(1,-2)$, $a=2>0$ (opens up)

Step2: Match to graph

Matches Graph C
---

Problem 36: $g(x)=\frac{1}{2}(x+1)^2-2$

Step1: Identify vertex and direction

Vertex $(-1,-2)$, $a=\frac{1}{2}>0$ (opens up, wide)

Step2: Match to graph

Matches Graph B
---

Problem 37: $g(x)=-2(x-1)^2+2$

Step1: Identify vertex and direction

Vertex $(1,2)$, $a=-2<0$ (opens down)

Step2: Match to graph

Matches Graph D
---

Problem 38: $g(x)=2(x+1)^2+2$

Step1: Identify vertex and direction

Vertex $(-1,2)$, $a=2>0$ (opens up)

Step2: Match to graph

Matches Graph E
---

Problem 39: $g(x)=-2(x+1)^2-2$

Step1: Identify vertex and direction

Vertex $(-1,-2)$, $a=-2<0$ (opens down)

Step2: Match to graph

Matches Graph F
---

Problem 40: $g(x)=2(x-1)^2+2$

Step1: Identify vertex and direction

Vertex $(1,2)$, $a=2>0$ (opens up)

Step2: Match to graph

Matches Graph A

Answer:

1. Graph C
2. Graph B
3. Graph D
4. Graph E
5. Graph F
6. Graph A