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Sagot :
Let's analyze each of the statements about the function [tex]\( f(x) = -x^2 - 4x + 2 \)[/tex].
1. The domain is [tex]\(\{x \mid x \leq -2\}\)[/tex].
This statement is false. The domain of any polynomial function, including this quadratic function, is all real numbers. It is not limited to [tex]\( x \leq -2 \)[/tex].
2. The range is [tex]\(\{y \mid y \leq 6\}\)[/tex].
This statement is true. To determine the range, we need to find the vertex of the quadratic function. The vertex form of a parabola [tex]\( y = ax^2 + bx + c \)[/tex] can be found using [tex]\(-\frac{b}{2a}\)[/tex] for the x-coordinate. In this function, [tex]\( a = -1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = 2 \)[/tex].
- Calculate the x-coordinate of the vertex:
[tex]\[ x = -\frac{-4}{2 \cdot -1} = \frac{4}{-2} = -2 \][/tex]
- Calculate the y-coordinate of the vertex by substituting [tex]\( x = -2 \)[/tex] into the function:
[tex]\[ y = -(-2)^2 - 4(-2) + 2 = -4 + 8 + 2 = 6 \][/tex]
Since the parabola opens downward (coefficient of [tex]\( x^2 \)[/tex] is negative), the maximum y-value (vertex y-coordinate) is 6. Therefore, the range is [tex]\( \{y \mid y \leq 6\} \)[/tex].
3. The function is increasing over the interval [tex]\((-\infty, -2)\)[/tex].
This statement is true. A downward-opening parabola is increasing to the left of its vertex. The vertex is at [tex]\( x = -2 \)[/tex], so the function is increasing over the interval [tex]\((-\infty, -2)\)[/tex].
4. The function is decreasing over the interval [tex]\((-4, \infty)\)[/tex].
This statement is false. The function decreases to the right of the vertex. The vertex is at [tex]\( x = -2 \)[/tex], so the function is decreasing over the interval [tex]\((-2, \infty)\)[/tex], not [tex]\((-4, \infty)\)[/tex].
5. The function has a positive [tex]\( y \)[/tex]-intercept.
This statement is true. The y-intercept occurs when [tex]\( x = 0 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ y = -0^2 - 4(0) + 2 = 2 \][/tex]
The y-intercept is 2, which is positive.
So, to summarize:
- Statement 1 is false.
- Statement 2 is true.
- Statement 3 is true.
- Statement 4 is false.
- Statement 5 is true.
1. The domain is [tex]\(\{x \mid x \leq -2\}\)[/tex].
This statement is false. The domain of any polynomial function, including this quadratic function, is all real numbers. It is not limited to [tex]\( x \leq -2 \)[/tex].
2. The range is [tex]\(\{y \mid y \leq 6\}\)[/tex].
This statement is true. To determine the range, we need to find the vertex of the quadratic function. The vertex form of a parabola [tex]\( y = ax^2 + bx + c \)[/tex] can be found using [tex]\(-\frac{b}{2a}\)[/tex] for the x-coordinate. In this function, [tex]\( a = -1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = 2 \)[/tex].
- Calculate the x-coordinate of the vertex:
[tex]\[ x = -\frac{-4}{2 \cdot -1} = \frac{4}{-2} = -2 \][/tex]
- Calculate the y-coordinate of the vertex by substituting [tex]\( x = -2 \)[/tex] into the function:
[tex]\[ y = -(-2)^2 - 4(-2) + 2 = -4 + 8 + 2 = 6 \][/tex]
Since the parabola opens downward (coefficient of [tex]\( x^2 \)[/tex] is negative), the maximum y-value (vertex y-coordinate) is 6. Therefore, the range is [tex]\( \{y \mid y \leq 6\} \)[/tex].
3. The function is increasing over the interval [tex]\((-\infty, -2)\)[/tex].
This statement is true. A downward-opening parabola is increasing to the left of its vertex. The vertex is at [tex]\( x = -2 \)[/tex], so the function is increasing over the interval [tex]\((-\infty, -2)\)[/tex].
4. The function is decreasing over the interval [tex]\((-4, \infty)\)[/tex].
This statement is false. The function decreases to the right of the vertex. The vertex is at [tex]\( x = -2 \)[/tex], so the function is decreasing over the interval [tex]\((-2, \infty)\)[/tex], not [tex]\((-4, \infty)\)[/tex].
5. The function has a positive [tex]\( y \)[/tex]-intercept.
This statement is true. The y-intercept occurs when [tex]\( x = 0 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ y = -0^2 - 4(0) + 2 = 2 \][/tex]
The y-intercept is 2, which is positive.
So, to summarize:
- Statement 1 is false.
- Statement 2 is true.
- Statement 3 is true.
- Statement 4 is false.
- Statement 5 is true.
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