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Let's analyze each of the given functions to identify which are parabolas and their orientation (facing up or down).
### Parabolas
A parabola is represented by a quadratic function of the form [tex]\( y = ax^2 + bx + c \)[/tex]. The orientation of the parabola is determined by the sign of the coefficient [tex]\( a \)[/tex]:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards (facing up).
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards (facing down).
### Linear Functions
Linear functions are of the form [tex]\( y = mx + b \)[/tex]. These are not parabolas.
### Identifying and Categorizing the Functions
1. [tex]\( y = (x-3)(2x-4) \)[/tex]
- Expanding this:
[tex]\[ y = x(2x - 4) - 3(2x - 4) = 2x^2 - 4x - 6x + 12 = 2x^2 - 10x + 12 \][/tex]
- Here, [tex]\( a = 2 > 0 \)[/tex]. Therefore, it is a parabola facing up.
2. [tex]\( y = 2x + 5 \)[/tex]
- This is a linear function and not a parabola.
3. [tex]\( y = -2x^2 + x - 5 \)[/tex]
- Here, [tex]\( a = -2 < 0 \)[/tex]. Therefore, it is a parabola facing down.
4. [tex]\( y = 7(x+2)^2 - 9 \)[/tex]
- Expanding this:
[tex]\[ y = 7(x^2 + 4x + 4) - 9 = 7x^2 + 28x + 28 - 9 = 7x^2 + 28x + 19 \][/tex]
- Here, [tex]\( a = 7 > 0 \)[/tex]. Therefore, it is a parabola facing up.
5. [tex]\( y = (2-x)(x+5) \)[/tex]
- Expanding this:
[tex]\[ y = 2x + 10 - x^2 - 5 = -x^2 + 2x + 10 - 5 = -x^2 + 2x + 5 \][/tex]
- Here, [tex]\( a = -1 < 0 \)[/tex]. Therefore, it is a parabola facing down.
6. [tex]\( y = -3x + 8 \)[/tex]
- This is a linear function and not a parabola.
7. [tex]\( y = 3x^2 - 2x + 1 \)[/tex]
- Here, [tex]\( a = 3 > 0 \)[/tex]. Therefore, it is a parabola facing up.
8. [tex]\( y = -(x+5)^2 + 3 \)[/tex]
- Expanding this:
[tex]\[ y = -(x^2 + 10x + 25) + 3 = -x^2 - 10x - 25 + 3 = -x^2 - 10x - 22 \][/tex]
- Here, [tex]\( a = -1 < 0 \)[/tex]. Therefore, it is a parabola facing down.
### Final Categorization
#### Parabolas facing up:
1. [tex]\( y = (x-3)(2x-4) \)[/tex]
2. [tex]\( y = 7(x+2)^2 - 9 \)[/tex]
3. [tex]\( y = 3x^2 - 2x + 1 \)[/tex]
#### Parabolas facing down:
1. [tex]\( y = -2x^2 + x - 5 \)[/tex]
2. [tex]\( y = (2-x)(x+5) \)[/tex]
3. [tex]\( y = -(x+5)^2 + 3 \)[/tex]
### Parabolas
A parabola is represented by a quadratic function of the form [tex]\( y = ax^2 + bx + c \)[/tex]. The orientation of the parabola is determined by the sign of the coefficient [tex]\( a \)[/tex]:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards (facing up).
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards (facing down).
### Linear Functions
Linear functions are of the form [tex]\( y = mx + b \)[/tex]. These are not parabolas.
### Identifying and Categorizing the Functions
1. [tex]\( y = (x-3)(2x-4) \)[/tex]
- Expanding this:
[tex]\[ y = x(2x - 4) - 3(2x - 4) = 2x^2 - 4x - 6x + 12 = 2x^2 - 10x + 12 \][/tex]
- Here, [tex]\( a = 2 > 0 \)[/tex]. Therefore, it is a parabola facing up.
2. [tex]\( y = 2x + 5 \)[/tex]
- This is a linear function and not a parabola.
3. [tex]\( y = -2x^2 + x - 5 \)[/tex]
- Here, [tex]\( a = -2 < 0 \)[/tex]. Therefore, it is a parabola facing down.
4. [tex]\( y = 7(x+2)^2 - 9 \)[/tex]
- Expanding this:
[tex]\[ y = 7(x^2 + 4x + 4) - 9 = 7x^2 + 28x + 28 - 9 = 7x^2 + 28x + 19 \][/tex]
- Here, [tex]\( a = 7 > 0 \)[/tex]. Therefore, it is a parabola facing up.
5. [tex]\( y = (2-x)(x+5) \)[/tex]
- Expanding this:
[tex]\[ y = 2x + 10 - x^2 - 5 = -x^2 + 2x + 10 - 5 = -x^2 + 2x + 5 \][/tex]
- Here, [tex]\( a = -1 < 0 \)[/tex]. Therefore, it is a parabola facing down.
6. [tex]\( y = -3x + 8 \)[/tex]
- This is a linear function and not a parabola.
7. [tex]\( y = 3x^2 - 2x + 1 \)[/tex]
- Here, [tex]\( a = 3 > 0 \)[/tex]. Therefore, it is a parabola facing up.
8. [tex]\( y = -(x+5)^2 + 3 \)[/tex]
- Expanding this:
[tex]\[ y = -(x^2 + 10x + 25) + 3 = -x^2 - 10x - 25 + 3 = -x^2 - 10x - 22 \][/tex]
- Here, [tex]\( a = -1 < 0 \)[/tex]. Therefore, it is a parabola facing down.
### Final Categorization
#### Parabolas facing up:
1. [tex]\( y = (x-3)(2x-4) \)[/tex]
2. [tex]\( y = 7(x+2)^2 - 9 \)[/tex]
3. [tex]\( y = 3x^2 - 2x + 1 \)[/tex]
#### Parabolas facing down:
1. [tex]\( y = -2x^2 + x - 5 \)[/tex]
2. [tex]\( y = (2-x)(x+5) \)[/tex]
3. [tex]\( y = -(x+5)^2 + 3 \)[/tex]
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