Asked by Tesla
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.
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Answer: The axis of symmetry is x = 0.
Reason: For f(x) = ax^2 + bx + c the vertex x-coordinate is h = -b/(2a), and the y-intercept is at x = 0. If the vertex equals the y-intercept then h = 0, so -b/(2a) = 0 ⇒ b = 0. Thus f(x) = ax^2 + c, which is symmetric about the vertical line x = 0.
The axis y = 0 is the horizontal axis and is not the axis of symmetry. The number of x-intercepts depends on a and c (none, one, or two), so neither of the intercept statements must be true.
Reason: For f(x) = ax^2 + bx + c the vertex x-coordinate is h = -b/(2a), and the y-intercept is at x = 0. If the vertex equals the y-intercept then h = 0, so -b/(2a) = 0 ⇒ b = 0. Thus f(x) = ax^2 + c, which is symmetric about the vertical line x = 0.
The axis y = 0 is the horizontal axis and is not the axis of symmetry. The number of x-intercepts depends on a and c (none, one, or two), so neither of the intercept statements must be true.
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