Question
Which must be true of a quadratic function whose vertex is the same as its y-intercept?
The axis of symmetry for the function is x = 0.
The axis of symmetry for the function is y = 0.
The function has no x-intercepts.
The function has 1 x-intercept.
The axis of symmetry for the function is x = 0.
The axis of symmetry for the function is y = 0.
The function has no x-intercepts.
The function has 1 x-intercept.
Answers
GPT-4o mini
If a quadratic function has its vertex at the same point as its y-intercept, we can deduce several things about the function's characteristics.
1. The y-intercept occurs when \( x = 0 \), and for the vertex to also be at this point, it follows that the vertex form of the quadratic function can be expressed as \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. Since the vertex is the same as the y-intercept, we have \( h = 0 \) and \( k = f(0) \). Hence, the vertex is \( (0, f(0)) \).
2. The axis of symmetry of a parabola described by its vertex form is \( x = h \). Since \( h = 0 \) in this case, the axis of symmetry is \( x = 0 \).
3. A quadratic function can have 0, 1, or 2 x-intercepts. If the vertex is on the y-axis and represents the lowest point (in case of \( a > 0 \)) or the highest point (in case of \( a < 0 \)), it will have one x-intercept if \( k = 0 \), and no x-intercepts if \( k > 0 \) (for a minimum) or \( k < 0 \) (for a maximum).
Given this analysis, we conclude that the statement which must be true for the quadratic function whose vertex is the same as its y-intercept is:
**The axis of symmetry for the function is \( x = 0 \).**
The other statements may or may not be true depending on the specific value of \( k \).
1. The y-intercept occurs when \( x = 0 \), and for the vertex to also be at this point, it follows that the vertex form of the quadratic function can be expressed as \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. Since the vertex is the same as the y-intercept, we have \( h = 0 \) and \( k = f(0) \). Hence, the vertex is \( (0, f(0)) \).
2. The axis of symmetry of a parabola described by its vertex form is \( x = h \). Since \( h = 0 \) in this case, the axis of symmetry is \( x = 0 \).
3. A quadratic function can have 0, 1, or 2 x-intercepts. If the vertex is on the y-axis and represents the lowest point (in case of \( a > 0 \)) or the highest point (in case of \( a < 0 \)), it will have one x-intercept if \( k = 0 \), and no x-intercepts if \( k > 0 \) (for a minimum) or \( k < 0 \) (for a maximum).
Given this analysis, we conclude that the statement which must be true for the quadratic function whose vertex is the same as its y-intercept is:
**The axis of symmetry for the function is \( x = 0 \).**
The other statements may or may not be true depending on the specific value of \( k \).