Question
The quadratic function can be rewritten as both and Which statement is true about the quadratic function?
The graph of the function has a minimum at ( 1 , 16 ) .
The graph of the function has a y − intercept at ( 0 , 12 ) .
The line of symmetry of the function is x = − 1.
The function has zeros at ( − 3 , 0 ) and ( 1 , 0 ) .
The graph of the function has a minimum at ( 1 , 16 ) .
The graph of the function has a y − intercept at ( 0 , 12 ) .
The line of symmetry of the function is x = − 1.
The function has zeros at ( − 3 , 0 ) and ( 1 , 0 ) .
Answers
GPT-4o mini
To determine which statement is true about the quadratic function, we should analyze the provided options for validity. However, the specific quadratic function was not explicitly provided in your statement. Typically, a quadratic function can be represented in standard form \( f(x) = ax^2 + bx + c \) or in vertex form \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.
**Let's analyze each statement:**
1. **The graph of the function has a minimum at (1, 16).**
- This statement implies that the vertex of the quadratic function is at \( x = 1 \) and \( y = 16 \). If this were true, then the quadratic would open upwards and would indeed have a minimum at this point.
2. **The graph of the function has a y-intercept at (0, 12).**
- The y-intercept occurs when \( x = 0 \). To verify this, we would need to calculate \( f(0) \) using the given quadratic function, which is not provided. If \( f(0) = 12 \), then this statement is true.
3. **The line of symmetry of the function is \( x = -1 \).**
- The line of symmetry of a quadratic function represented in standard form \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \). We would need the coefficients \( a \) and \( b \) to verify this.
4. **The function has zeros at (-3, 0) and (1, 0).**
- Zeros (or x-intercepts) are the points where the function equals zero. This means the quadratic can be factored to find these points. If \( -3 \) and \( 1 \) are indeed roots, then the function could be represented as \( f(x) = a(x + 3)(x - 1) \) for some \( a \) (where \( a \) is not zero).
**Conclusion:**
To find the truth of the statements provided, we would need the actual quadratic function. Once you find it, calculate the vertex to determine minimum points, substitute to find the y-intercept, find the coefficients for the line of symmetry, and check the roots for the location of zeros.
Please provide the specific quadratic function so these statements can be analyzed more thoroughly for precision.
**Let's analyze each statement:**
1. **The graph of the function has a minimum at (1, 16).**
- This statement implies that the vertex of the quadratic function is at \( x = 1 \) and \( y = 16 \). If this were true, then the quadratic would open upwards and would indeed have a minimum at this point.
2. **The graph of the function has a y-intercept at (0, 12).**
- The y-intercept occurs when \( x = 0 \). To verify this, we would need to calculate \( f(0) \) using the given quadratic function, which is not provided. If \( f(0) = 12 \), then this statement is true.
3. **The line of symmetry of the function is \( x = -1 \).**
- The line of symmetry of a quadratic function represented in standard form \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \). We would need the coefficients \( a \) and \( b \) to verify this.
4. **The function has zeros at (-3, 0) and (1, 0).**
- Zeros (or x-intercepts) are the points where the function equals zero. This means the quadratic can be factored to find these points. If \( -3 \) and \( 1 \) are indeed roots, then the function could be represented as \( f(x) = a(x + 3)(x - 1) \) for some \( a \) (where \( a \) is not zero).
**Conclusion:**
To find the truth of the statements provided, we would need the actual quadratic function. Once you find it, calculate the vertex to determine minimum points, substitute to find the y-intercept, find the coefficients for the line of symmetry, and check the roots for the location of zeros.
Please provide the specific quadratic function so these statements can be analyzed more thoroughly for precision.