Question
the quadratic function f (x) = x^2 + 2x - 15 is showen in the graph. determine the domain and range of this function.
a domain R: range y > -16
b domain R: range y > 16
c domain R: range y < -16
d domain R: range R
a domain R: range y > -16
b domain R: range y > 16
c domain R: range y < -16
d domain R: range R
Answers
Answered by
GPT-4o mini
To determine the domain and range of the quadratic function \( f(x) = x^2 + 2x - 15 \), let's analyze it step by step.
1. **Domain**:
The domain of a quadratic function is always all real numbers, \( R \). This is because you can input any real number into the function without encountering any restrictions such as division by zero or taking the square root of a negative number.
2. **Range**:
To find the range, we first need to find the vertex of the quadratic function, as it will determine the minimum or maximum value of the function. The standard form of a quadratic function is \( ax^2 + bx + c \), where \( a = 1 \), \( b = 2 \), and \( c = -15 \).
The x-coordinate of the vertex is given by the formula:
\[
x = -\frac{b}{2a} = -\frac{2}{2 \times 1} = -1
\]
Now, we can find the y-coordinate of the vertex by substituting \( x = -1 \) into the function:
\[
f(-1) = (-1)^2 + 2(-1) - 15 = 1 - 2 - 15 = -16
\]
Since the coefficient of \( x^2 \) (which is \( a = 1 \)) is positive, this means the parabola opens upwards and the vertex represents the minimum point on the graph. Therefore, the range of the function will be all real numbers greater than or equal to the y-coordinate of the vertex.
Thus, the range is:
\[
y \geq -16
\]
Given this analysis, the correct answer is:
- **Domain**: \( R \)
- **Range**: \( y \geq -16 \)
So the correct option is:
a) domain \( R \): range \( y \geq -16 \) (note: there seems to be a little mistake in your earlier phrasing "y > -16", which should be corrected to "y ≥ -16").
1. **Domain**:
The domain of a quadratic function is always all real numbers, \( R \). This is because you can input any real number into the function without encountering any restrictions such as division by zero or taking the square root of a negative number.
2. **Range**:
To find the range, we first need to find the vertex of the quadratic function, as it will determine the minimum or maximum value of the function. The standard form of a quadratic function is \( ax^2 + bx + c \), where \( a = 1 \), \( b = 2 \), and \( c = -15 \).
The x-coordinate of the vertex is given by the formula:
\[
x = -\frac{b}{2a} = -\frac{2}{2 \times 1} = -1
\]
Now, we can find the y-coordinate of the vertex by substituting \( x = -1 \) into the function:
\[
f(-1) = (-1)^2 + 2(-1) - 15 = 1 - 2 - 15 = -16
\]
Since the coefficient of \( x^2 \) (which is \( a = 1 \)) is positive, this means the parabola opens upwards and the vertex represents the minimum point on the graph. Therefore, the range of the function will be all real numbers greater than or equal to the y-coordinate of the vertex.
Thus, the range is:
\[
y \geq -16
\]
Given this analysis, the correct answer is:
- **Domain**: \( R \)
- **Range**: \( y \geq -16 \)
So the correct option is:
a) domain \( R \): range \( y \geq -16 \) (note: there seems to be a little mistake in your earlier phrasing "y > -16", which should be corrected to "y ≥ -16").