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To determine the range of Ethan's function [tex]\( f(x) = 2x^2 + 2x + 4 \)[/tex], we need to analyze its properties, specifically focusing on the minimum value of the quadratic function and how the function behaves.
### Step 1: Identify the Components of the Function
The function [tex]\( f(x) = 2x^2 + 2x + 4 \)[/tex] is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 2 \)[/tex]
- [tex]\( c = 4 \)[/tex]
### Step 2: Determine the Vertex of the Quadratic Function
For any quadratic function [tex]\( ax^2 + bx + c \)[/tex], the vertex, which gives the minimum value of the function (since the parabola opens upwards when [tex]\( a > 0 \)[/tex]), occurs at [tex]\( x = -\frac{b}{2a} \)[/tex].
So, let's plug in our values:
[tex]\[ x = -\frac{2}{2 \cdot 2} = -\frac{2}{4} = -0.5 \][/tex]
### Step 3: Calculate the Minimum Value of the Function at the Vertex
We now substitute [tex]\( x = -0.5 \)[/tex] back into the function [tex]\( f(x) = 2x^2 + 2x + 4 \)[/tex] to find the minimum value of [tex]\( f(x) \)[/tex]:
[tex]\[ f(-0.5) = 2(-0.5)^2 + 2(-0.5) + 4 \][/tex]
[tex]\[ f(-0.5) = 2(0.25) + 2(-0.5) + 4 \][/tex]
[tex]\[ f(-0.5) = 0.5 - 1 + 4 \][/tex]
[tex]\[ f(-0.5) = 3.5 \][/tex]
### Step 4: Determine the Range of the Function
Since the function [tex]\( f(x) = 2x^2 + 2x + 4 \)[/tex] is a quadratic function that opens upwards (the coefficient of [tex]\( x^2 \)[/tex], [tex]\( a = 2 \)[/tex], is positive), the range of the function is all values of [tex]\( f(x) \)[/tex] starting from the minimum value upwards to infinity.
Thus, the minimum value of [tex]\( f(x) \)[/tex] is [tex]\( f(-0.5) = 3.5 \)[/tex].
### Step 5: State the Range
Therefore, the range of Ethan's function [tex]\( f(x) = 2x^2 + 2x + 4 \)[/tex] is:
[tex]\[ [3.5, \infty) \][/tex]
### Conclusion
So, the correct statement about the range of Ethan's function is:
[tex]\[ \boxed{\text{The range of Ethan's function is } [3.5, \infty)} \][/tex]
### Step 1: Identify the Components of the Function
The function [tex]\( f(x) = 2x^2 + 2x + 4 \)[/tex] is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 2 \)[/tex]
- [tex]\( c = 4 \)[/tex]
### Step 2: Determine the Vertex of the Quadratic Function
For any quadratic function [tex]\( ax^2 + bx + c \)[/tex], the vertex, which gives the minimum value of the function (since the parabola opens upwards when [tex]\( a > 0 \)[/tex]), occurs at [tex]\( x = -\frac{b}{2a} \)[/tex].
So, let's plug in our values:
[tex]\[ x = -\frac{2}{2 \cdot 2} = -\frac{2}{4} = -0.5 \][/tex]
### Step 3: Calculate the Minimum Value of the Function at the Vertex
We now substitute [tex]\( x = -0.5 \)[/tex] back into the function [tex]\( f(x) = 2x^2 + 2x + 4 \)[/tex] to find the minimum value of [tex]\( f(x) \)[/tex]:
[tex]\[ f(-0.5) = 2(-0.5)^2 + 2(-0.5) + 4 \][/tex]
[tex]\[ f(-0.5) = 2(0.25) + 2(-0.5) + 4 \][/tex]
[tex]\[ f(-0.5) = 0.5 - 1 + 4 \][/tex]
[tex]\[ f(-0.5) = 3.5 \][/tex]
### Step 4: Determine the Range of the Function
Since the function [tex]\( f(x) = 2x^2 + 2x + 4 \)[/tex] is a quadratic function that opens upwards (the coefficient of [tex]\( x^2 \)[/tex], [tex]\( a = 2 \)[/tex], is positive), the range of the function is all values of [tex]\( f(x) \)[/tex] starting from the minimum value upwards to infinity.
Thus, the minimum value of [tex]\( f(x) \)[/tex] is [tex]\( f(-0.5) = 3.5 \)[/tex].
### Step 5: State the Range
Therefore, the range of Ethan's function [tex]\( f(x) = 2x^2 + 2x + 4 \)[/tex] is:
[tex]\[ [3.5, \infty) \][/tex]
### Conclusion
So, the correct statement about the range of Ethan's function is:
[tex]\[ \boxed{\text{The range of Ethan's function is } [3.5, \infty)} \][/tex]
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