Answer :
Sure, let's solve the problems given the quadratic equation \( y = x^2 + 2x - 8 \).
### 1a) Does the parabola open up or down?
The general form for a quadratic equation is \( y = ax^2 + bx + c \).
In our equation, \( y = x^2 + 2x - 8 \):
- The coefficient \( a \) of \( x^2 \) is 1.
- The coefficient \( b \) of \( x \) is 2.
- The constant term \( c \) is -8.
To determine whether the parabola opens up or down, we look at the coefficient \( a \):
- If \( a > 0 \), the parabola opens up.
- If \( a < 0 \), the parabola opens down.
Here, \( a = 1 \), which is greater than 0. Therefore, the parabola opens up.
### 1b) Find the vertex using the formula \( x = \frac{-b}{2a} \):
The vertex form of a parabola can be found by using the formula for the x-coordinate of the vertex:
[tex]\[ x = \frac{-b}{2a} \][/tex]
For our equation \( y = x^2 + 2x - 8 \):
- \( a = 1 \)
- \( b = 2 \)
Plugging these values into the formula:
[tex]\[ x = \frac{-2}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-2}{2} \][/tex]
[tex]\[ x = -1 \][/tex]
Now we need to find the y-coordinate by substituting \( x = -1 \) back into the original equation:
[tex]\[ y = (-1)^2 + 2(-1) - 8 \][/tex]
[tex]\[ y = 1 - 2 - 8 \][/tex]
[tex]\[ y = -9 \][/tex]
So, the vertex of the parabola is:
[tex]\[ (-1, -9) \][/tex]
### Summary:
1. The parabola opens up.
2. The vertex is at the point (-1, -9), found by calculating [tex]\(\ x = \frac{-b}{2a}\)[/tex] and substituting back to find the y-value.
### 1a) Does the parabola open up or down?
The general form for a quadratic equation is \( y = ax^2 + bx + c \).
In our equation, \( y = x^2 + 2x - 8 \):
- The coefficient \( a \) of \( x^2 \) is 1.
- The coefficient \( b \) of \( x \) is 2.
- The constant term \( c \) is -8.
To determine whether the parabola opens up or down, we look at the coefficient \( a \):
- If \( a > 0 \), the parabola opens up.
- If \( a < 0 \), the parabola opens down.
Here, \( a = 1 \), which is greater than 0. Therefore, the parabola opens up.
### 1b) Find the vertex using the formula \( x = \frac{-b}{2a} \):
The vertex form of a parabola can be found by using the formula for the x-coordinate of the vertex:
[tex]\[ x = \frac{-b}{2a} \][/tex]
For our equation \( y = x^2 + 2x - 8 \):
- \( a = 1 \)
- \( b = 2 \)
Plugging these values into the formula:
[tex]\[ x = \frac{-2}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-2}{2} \][/tex]
[tex]\[ x = -1 \][/tex]
Now we need to find the y-coordinate by substituting \( x = -1 \) back into the original equation:
[tex]\[ y = (-1)^2 + 2(-1) - 8 \][/tex]
[tex]\[ y = 1 - 2 - 8 \][/tex]
[tex]\[ y = -9 \][/tex]
So, the vertex of the parabola is:
[tex]\[ (-1, -9) \][/tex]
### Summary:
1. The parabola opens up.
2. The vertex is at the point (-1, -9), found by calculating [tex]\(\ x = \frac{-b}{2a}\)[/tex] and substituting back to find the y-value.
