Example Question - vertex form of parabola
Here are examples of questions we've helped users solve.
Finding the Equation of a Parabola with Given Parameters
To find the equation of a parabola that opens upwards, has a minimum value of -3, and an axis of symmetry at x = -3, you should look for a quadratic equation in the vertex form: \( f(x) = a(x-h)^2 + k \) where (h, k) is the vertex of the parabola. Since the axis of symmetry is at x = -3, h must be -3. The minimum value of -3 indicates that k = -3, and since the parabola opens upwards, the coefficient 'a' must be positive. Looking at the options: A. \( f(x) = -(x+3)^2 - 3 \) - This option has both a negative 'a' (which would open downwards) and the wrong sign in front of 3 (it would result in an axis of symmetry at x = -(-3) = 3, not x = -3). B. \( f(x) = (x+3)^2 - 6 \) - This equation has the right sign for 'a' (which would open upwards) but its minimum value would be -6, not -3. C. \( f(x) = (x-3)^2 + 6 \) - This equation has the axis of symmetry at x = 3, not x = -3. Additionally, its vertex would be at (3, 6), which does not match the given minimum value of -3. D. \( f(x) = (x+3)^2 - 3 \) - This is the correct equation. It has a positive 'a', indicating that the parabola opens upwards, and the vertex form matches the given vertex of the parabola: (h, k) = (-3, -3). Therefore, the correct equation is: D. \( f(x) = (x+3)^2 - 3 \)
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.
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