Answer :
To determine the correct statements about the graph of the inequality \( y < x^2 + 2x - 6 \), we need to analyze the properties of the quadratic function \( y = x^2 + 2x - 6 \). Here's a step-by-step breakdown:
### 1. Direction of the Parabola:
The coefficient of \( x^2 \) in the quadratic function is +1, which is positive. Therefore, the parabola opens up.
### 2. Shading:
The inequality given is \( y < x^2 + 2x - 6 \). In this case, the shading is below or outside the parabola since it represents all the points where \( y \) is less than the expression \( x^2 + 2x - 6 \).
### 3. Vertex of the Parabola:
To find the vertex of the parabola \( y = x^2 + 2x - 6 \), we use the vertex formula \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients of \( x^2 \) and \( x \), respectively. Here:
- \( a = 1 \)
- \( b = 2 \)
So, the \( x \)-coordinate of the vertex is:
[tex]\[ x = -\frac{2}{2 \cdot 1} = -1 \][/tex]
To find the \( y \)-coordinate of the vertex, substitute \( x = -1 \) back into the equation \( y = x^2 + 2x - 6 \):
[tex]\[ y = (-1)^2 + 2(-1) - 6 \][/tex]
[tex]\[ y = 1 - 2 - 6 \][/tex]
[tex]\[ y = -7 \][/tex]
Thus, the vertex is at \((-1, -7)\).
### 4. Nature of the Line:
The given inequality \( y < x^2 + 2x - 6 \) is a strict inequality (less than, but not equal to). This means that the boundary line of the parabola is dashed to indicate that points on the line \( y = x^2 + 2x - 6 \) are not included in the solution set.
### Summary:
Based on our analysis:
- The parabola does not open down; it opens up.
- The shading is below or outside the parabola.
- The vertex of the parabola is correctly located at \( (-1, -7) \).
- The parabola is represented by a dashed line, not a solid line.
Therefore, the true statement about the graph of \( y < x^2 + 2x - 6 \) is:
- The vertex is located at [tex]\( (-1, -7) \)[/tex].
### 1. Direction of the Parabola:
The coefficient of \( x^2 \) in the quadratic function is +1, which is positive. Therefore, the parabola opens up.
### 2. Shading:
The inequality given is \( y < x^2 + 2x - 6 \). In this case, the shading is below or outside the parabola since it represents all the points where \( y \) is less than the expression \( x^2 + 2x - 6 \).
### 3. Vertex of the Parabola:
To find the vertex of the parabola \( y = x^2 + 2x - 6 \), we use the vertex formula \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients of \( x^2 \) and \( x \), respectively. Here:
- \( a = 1 \)
- \( b = 2 \)
So, the \( x \)-coordinate of the vertex is:
[tex]\[ x = -\frac{2}{2 \cdot 1} = -1 \][/tex]
To find the \( y \)-coordinate of the vertex, substitute \( x = -1 \) back into the equation \( y = x^2 + 2x - 6 \):
[tex]\[ y = (-1)^2 + 2(-1) - 6 \][/tex]
[tex]\[ y = 1 - 2 - 6 \][/tex]
[tex]\[ y = -7 \][/tex]
Thus, the vertex is at \((-1, -7)\).
### 4. Nature of the Line:
The given inequality \( y < x^2 + 2x - 6 \) is a strict inequality (less than, but not equal to). This means that the boundary line of the parabola is dashed to indicate that points on the line \( y = x^2 + 2x - 6 \) are not included in the solution set.
### Summary:
Based on our analysis:
- The parabola does not open down; it opens up.
- The shading is below or outside the parabola.
- The vertex of the parabola is correctly located at \( (-1, -7) \).
- The parabola is represented by a dashed line, not a solid line.
Therefore, the true statement about the graph of \( y < x^2 + 2x - 6 \) is:
- The vertex is located at [tex]\( (-1, -7) \)[/tex].
