6. Given the quadratic function shown below, which of the following gives a domain interval over which this function is only decreasing? (1) \( x>-4 \) (2) \( x<-4 \) (3) \( x>2 \) (4) \( x<2 \)

To determine the domain interval over which the given quadratic function is only decreasing, we need to analyze the properties of quadratic functions. 1. **Understanding Quadratic Functions**: A quadratic function can be expressed in the standard form \( f(x) = ax^2 + bx + c \). The graph of a quadratic function is a parabola. If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards. 2. **Finding the Vertex**: The vertex of the parabola is the point where the function changes from increasing to decreasing (or vice versa). The x-coordinate of the vertex can be found using the formula: \[ x_v = -\frac{b}{2a} \] The function is decreasing to the left of the vertex and increasing to the right of the vertex. 3. **Analyzing the Options**: Without the specific coefficients of the quadratic function, we can still conclude that: - If the vertex \( x_v \) is at a certain point, the function will be decreasing for \( x < x_v \) and increasing for \( x > x_v \). 4. **Evaluating the Given Options**: - (1) \( x > -4 \): This suggests the function is increasing for values greater than -4. - (2) \( x < -4 \): This suggests the function is decreasing for values less than -4. - (3) \( x > 2 \): This suggests the function is increasing for values greater than 2. - (4) \( x < 2 \): This suggests the function is decreasing for values less than 2. 5. **Conclusion**: The correct answer will depend on the vertex of the quadratic function. However, if we assume the vertex is at \( x = -4 \) or \( x = 2 \), we can conclude: - If the vertex is at \( x = -4 \), the function is decreasing for \( x < -4 \). - If the vertex is at \( x = 2 \), the function is decreasing for \( x < 2 \). Since we are looking for intervals where the function is only decreasing, the options that indicate decreasing intervals are: - (2) \( x < -4 \) - (4) \( x < 2 \) Thus, the answer could be either (2) or (4) depending on the specific quadratic function. If we assume the vertex is at \( x = 2 \), then the best answer is (4) \( x < 2 \). If you have the specific quadratic function, please provide it for a more precise analysis.