Question
what is the rang of this quadratic function
y= x^2 + 9x + 14
y= x^2 + 9x + 14
Answers
GPT-4o mini
To find the range of the quadratic function \( y = x^2 + 9x + 14 \), we can follow these steps:
1. **Identify the coefficients**: The function is in the standard form of a quadratic equation \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = 9 \), and \( c = 14 \).
2. **Determine the direction of the parabola**: Since \( a = 1 \) is positive, the parabola opens upwards.
3. **Find the vertex**: The x-coordinate of the vertex of a quadratic function can be found using the formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{9}{2(1)} = -\frac{9}{2} = -4.5
\]
4. **Substitute the x-coordinate back into the function to find the y-coordinate of the vertex**:
\[
y = (-4.5)^2 + 9(-4.5) + 14
\]
\[
= 20.25 - 40.5 + 14
\]
\[
= 20.25 - 40.5 + 14 = -6.25
\]
5. **Determine the range**: Since the parabola opens upwards and the vertex is the lowest point (minimum value) of the function, the minimum value of \( y \) is \( -6.25 \). Therefore, the range of the function is all the values of \( y \) greater than or equal to \( -6.25 \).
Thus, the range of the quadratic function \( y = x^2 + 9x + 14 \) is:
\[
\text{Range} = [-6.25, \infty)
\]
1. **Identify the coefficients**: The function is in the standard form of a quadratic equation \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = 9 \), and \( c = 14 \).
2. **Determine the direction of the parabola**: Since \( a = 1 \) is positive, the parabola opens upwards.
3. **Find the vertex**: The x-coordinate of the vertex of a quadratic function can be found using the formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{9}{2(1)} = -\frac{9}{2} = -4.5
\]
4. **Substitute the x-coordinate back into the function to find the y-coordinate of the vertex**:
\[
y = (-4.5)^2 + 9(-4.5) + 14
\]
\[
= 20.25 - 40.5 + 14
\]
\[
= 20.25 - 40.5 + 14 = -6.25
\]
5. **Determine the range**: Since the parabola opens upwards and the vertex is the lowest point (minimum value) of the function, the minimum value of \( y \) is \( -6.25 \). Therefore, the range of the function is all the values of \( y \) greater than or equal to \( -6.25 \).
Thus, the range of the quadratic function \( y = x^2 + 9x + 14 \) is:
\[
\text{Range} = [-6.25, \infty)
\]