Question
The quadratic function can be rewritten as both and Which statement is true about the quadratic function?
Responses
The graph of the function has a minimum at
The graph of the function has a minimum at ( 1 , 16 ) .
The graph of the function has a intercept at
The graph of the function has a y − intercept at ( 0 , 12 ) .
The line of symmetry of the function is
The line of symmetry of the function is x = − 1.
The function has zeros at and
The function has zeros at ( − 3 , 0 ) and ( 1 , 0 ) .
Responses
The graph of the function has a minimum at
The graph of the function has a minimum at ( 1 , 16 ) .
The graph of the function has a intercept at
The graph of the function has a y − intercept at ( 0 , 12 ) .
The line of symmetry of the function is
The line of symmetry of the function is x = − 1.
The function has zeros at and
The function has zeros at ( − 3 , 0 ) and ( 1 , 0 ) .
Answers
Answered by
GPT-4o mini
To provide an accurate answer, we need to analyze the statements made about the quadratic function and what it implies. However, since specific details about the quadratic function were not given, I will explain how to evaluate each statement generally.
1. **Minimum Value**:
- If the quadratic function opens upwards (which is true if the coefficient of \( x^2 \) is positive), it will have a minimum, located at the vertex. The vertex can be found using the formula \( x = -\frac{b}{2a} \). You'll need to plug in the values of \( a \) and \( b \) to find the x-coordinate of the vertex, and then substitute back into the function to find the y-coordinate.
2. **Y-intercept**:
- The y-intercept of a quadratic function is found by evaluating the function at \( x = 0 \) (i.e., substitute \( x = 0 \) into the function). The y-coordinate of the intercept is the value of the function at that point.
3. **Line of Symmetry**:
- The line of symmetry for a quadratic function is always given by \( x = -\frac{b}{2a} \). This \( x \) value will be the same as the x-coordinate of the vertex.
4. **Zeros of the Function**:
- The zeros of the function (or the x-intercepts) can be determined by solving the quadratic equation \( ax^2 + bx + c = 0 \). You can use factoring, completing the square, or the quadratic formula to find the x-values where the function equals zero.
Now you can assess whether the statements match the derived information about your specific quadratic function.
If you find the following based on those evaluations:
- If the vertex calculation gives (1, 16), then "The graph of the function has a minimum at (1, 16)" is true.
- If the y-intercept calculation gives (0, 12), then "The graph of the function has a y − intercept at (0, 12)" is true.
- If the symmetry calculation gives \( x = -1 \), then "The line of symmetry of the function is x = -1" is true.
- If the zeros calculation gives (-3, 0) and (1, 0), then "The function has zeros at (-3, 0) and (1, 0)" is true.
Check each statement against your specific function to determine which is true.
1. **Minimum Value**:
- If the quadratic function opens upwards (which is true if the coefficient of \( x^2 \) is positive), it will have a minimum, located at the vertex. The vertex can be found using the formula \( x = -\frac{b}{2a} \). You'll need to plug in the values of \( a \) and \( b \) to find the x-coordinate of the vertex, and then substitute back into the function to find the y-coordinate.
2. **Y-intercept**:
- The y-intercept of a quadratic function is found by evaluating the function at \( x = 0 \) (i.e., substitute \( x = 0 \) into the function). The y-coordinate of the intercept is the value of the function at that point.
3. **Line of Symmetry**:
- The line of symmetry for a quadratic function is always given by \( x = -\frac{b}{2a} \). This \( x \) value will be the same as the x-coordinate of the vertex.
4. **Zeros of the Function**:
- The zeros of the function (or the x-intercepts) can be determined by solving the quadratic equation \( ax^2 + bx + c = 0 \). You can use factoring, completing the square, or the quadratic formula to find the x-values where the function equals zero.
Now you can assess whether the statements match the derived information about your specific quadratic function.
If you find the following based on those evaluations:
- If the vertex calculation gives (1, 16), then "The graph of the function has a minimum at (1, 16)" is true.
- If the y-intercept calculation gives (0, 12), then "The graph of the function has a y − intercept at (0, 12)" is true.
- If the symmetry calculation gives \( x = -1 \), then "The line of symmetry of the function is x = -1" is true.
- If the zeros calculation gives (-3, 0) and (1, 0), then "The function has zeros at (-3, 0) and (1, 0)" is true.
Check each statement against your specific function to determine which is true.