Example Question - equation 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 a Parabola with Specific Properties
The question in the image is asking for an equation of a parabola that opens upwards, has a minimum value of 8, and has an axis of symmetry at \(x = 3\). To find the correct equation, let's recall some properties of parabolas: 1. A parabola that opens upwards will have the form \(f(x) = a(x - h)^2 + k\), where \(a > 0\). 2. The vertex of the parabola is at the point \((h, k)\). This will also be the minimum point if the parabola opens upwards. 3. The axis of symmetry is the vertical line \(x = h\). Based on the given information, we need the parabola to open upwards which requires that \(a > 0\), its vertex will be at \((3, 8)\), meaning \(h = 3\) and \(k = 8\), and so the axis of symmetry is \(x = 3\). All the options in the image are in the vertex form of a parabola, \(f(x) = a(x - h)^2 + k\). We need to find the one that matches our vertex \((3, 8)\) and opens upwards. - Option A has \(h = 3\) and \(k = 8\) which is correct, and since \(a = 1\) (which is positive), this parabola opens upwards. This equation could be the correct answer. - Option B has the same \(h\) and \(k\), but a negative \(a\), meaning this parabola opens downwards. This cannot be the correct answer. - Option C has the correct \(k\), but \(h = -3\), which gives us the wrong axis of symmetry. So it's not correct. - Option D has \(h = 3\) and \(k = -8\), which means the vertex is at \((3, -8)\) and this does not match our minimum value of 8. Therefore, the correct answer is Option A: \(f(x) = (x - 3)^2 + 8\), because this is the only equation among the choices that correctly represents a parabola with the specified properties.
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.
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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