Question
What is the range of this quadratic function?
y=x2+2x–15
choices
{y|y ≤ -16}
{y|y ≥-16}
{y|y ≥-1}
all real numbers
Answers
GPT-4o mini
To find the range of the quadratic function \( y = x^2 + 2x - 15 \), we can first complete the square or find the vertex of the quadratic.
1. **Identify the coefficients**: The function is in the form \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = 2 \), and \( c = -15 \).
2. **Find the vertex**: The x-coordinate of the vertex for a quadratic function is given by the formula \( x = -\frac{b}{2a} \).
\[
x = -\frac{2}{2 \cdot 1} = -1
\]
3. **Find the y-coordinate of the vertex**: Substitute \( x = -1 \) back into the function to find the y-coordinate.
\[
y = (-1)^2 + 2(-1) - 15
\]
\[
y = 1 - 2 - 15
\]
\[
y = -16
\]
4. **Determine the direction of the parabola**: Since the coefficient of \( x^2 \) (which is 1) is positive, the parabola opens upwards. This means the vertex represents the minimum point of the function.
5. **Find the range**: Since the vertex is at \( (-1, -16) \) and the parabola opens upwards, the minimum value of \( y \) is -16, and it can take any value greater than or equal to -16.
So, the range of the function is \( \{ y | y \geq -16 \} \).
Therefore, the correct choice is:
**{y | y ≥ -16}**.
1. **Identify the coefficients**: The function is in the form \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = 2 \), and \( c = -15 \).
2. **Find the vertex**: The x-coordinate of the vertex for a quadratic function is given by the formula \( x = -\frac{b}{2a} \).
\[
x = -\frac{2}{2 \cdot 1} = -1
\]
3. **Find the y-coordinate of the vertex**: Substitute \( x = -1 \) back into the function to find the y-coordinate.
\[
y = (-1)^2 + 2(-1) - 15
\]
\[
y = 1 - 2 - 15
\]
\[
y = -16
\]
4. **Determine the direction of the parabola**: Since the coefficient of \( x^2 \) (which is 1) is positive, the parabola opens upwards. This means the vertex represents the minimum point of the function.
5. **Find the range**: Since the vertex is at \( (-1, -16) \) and the parabola opens upwards, the minimum value of \( y \) is -16, and it can take any value greater than or equal to -16.
So, the range of the function is \( \{ y | y \geq -16 \} \).
Therefore, the correct choice is:
**{y | y ≥ -16}**.