Example Question - upward-opening parabola
Here are examples of questions we've helped users solve.
Finding the Equation of an Upward-Opening Parabola
The question in the image is asking for an equation of a parabola that opens upward, has a minimum value of 3, and has an axis of symmetry at x = -3. A parabola that opens upward will have a positive coefficient for the squared term (x^2). The vertex form of a parabola's equation is: f(x) = a(x - h)^2 + k where (h, k) is the vertex of the parabola, and a determines the direction and width of the parabola. Here, we're looking for a parabola with a minimum value, which means its vertex will be at the bottom of the parabola (since it opens upward). The minimum value of the function is given by the value of k in the vertex form, which is 3 in this case. The axis of symmetry is x = h. For this problem, the axis of symmetry is given as x = -3, which means h must be -3. So we're looking for a parabola with a vertex at (-3, 3). Therefore, the equation would be in the form: f(x) = a(x + 3)^2 + 3 Since a must be positive (because the parabola opens upward), and none of the options in the image display a negative a, we can simply look for an equation that fits the vertex form with the correct h and k values. Option A matches the given requirements: A. f(x) = (x + 3)^2 + 3 Thus, the equation representing the parabola with the described properties is given by Option A.
Determining the Equation of an Upward-Opening Parabola with a Given Minimum Value and Axis of Symmetry
The question is asking for the equation of a parabola that opens upward, has a minimum value of 3, and an axis of symmetry at x=3. A parabola that opens upward must have a positive coefficient before the squared term in its equation. The minimum value of the parabola would be the y-coordinate of the vertex. Because we know the axis of symmetry is x = 3, this means the x-coordinate of the vertex is 3. The general form of a parabola's equation is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. If the parabola opens upwards and has a minimum value of 3, the k value (which represents the y-coordinate of the vertex) would be 3. Given this information, we can rule out option B and option C, because they have a minus sign before the squared term, which indicates the parabola opens downward, and because their k values are -6 and therefore can't represent a minimum value of 3. The only option that fits all criteria is option A: f(x) = (x - 3)^2 + 3, as this represents a parabola with the vertex at (3, 3), which means it opens upwards and has a minimum value of 3, and the axis of symmetry is at x = 3. Therefore, the correct answer is A.
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