Example Question - parabola characteristics
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Identifying a Parabola with Specific Characteristics
The question in the image asks which equation represents a parabola that opens upward, has a minimum value of 3, and has an axis of symmetry at \( x = -3 \). Parabolas that open upward have a positive coefficient in front of the \( x^2 \) term. A minimum value is represented by the vertex of the parabola, and in the vertex form of a parabola, \( y = a(x - h)^2 + k \), where the vertex is at the point \( (h, k) \), \( k \) will be the minimum value when the parabola opens upward. The axis of symmetry is at \( x = h \). Looking at the options given: A. \( f(x) = (x - (-3))^2 + 3 \) B. \( f(x) = -(x - (-3))^2 + 6 \) C. \( f(x) = (x - 3)^2 + 6 \) D. \( f(x) = (x - 3)^2 + 3 \) Option A, \( f(x) = (x - (-3))^2 + 3 \), simplifies to \( f(x) = (x + 3)^2 + 3 \), which has the correct axis of symmetry at \( x = -3 \) and a minimum value of 3. The coefficient in front of \( (x + 3)^2 \) is positive, indicating that the parabola opens upward. Options B, C, and D either do not have the correct axis of symmetry, or they have a negative leading coefficient (which would mean the parabola opens downward), or they don't have the correct minimum value. Therefore, the correct option is A. \( f(x) = (x + 3)^2 + 3 \).
Equation of a Parabola with Specific Characteristics
The question asks for the equation of a parabola that opens upwards, has a minimum value of 3, and an axis of symmetry at x=3. The general form of a parabola that opens upwards with a vertex at (h, k) is given by: f(x) = a(x - h)² + k where "h" is the x-coordinate of the vertex, "k" is the y-coordinate of the vertex (and the minimum value of the parabola, since it opens upwards), and "a" is a positive constant that affects the width of the parabola. Since the parabola opens upwards, a must be positive. Here, we are told the axis of symmetry is x=3, which means h is 3. Also, the parabola has a minimum value of 3, so k is also 3. Therefore, the equation becomes f(x) = a(x - 3)² + 3. The value of "a" is not specified, but any positive value of "a" would suffice for it to open upwards. The simplest form to choose is a=1, to match one of the given options. Looking at the options provided: A. f(x) = (x - 3)² + 3 (This matches our derived equation with a=1, h=3, and k=3) B. f(x) = (x - 3)² - 6 (This parabola also has an axis of symmetry at x=3, but it doesn't have a minimum value of 3, as required) C. f(x) = (x + 3)² - 6 (This one has an axis of symmetry at x=-3, which does not match our requirement) The correct answer is therefore: A. f(x) = (x - 3)² + 3
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