Example Question - axis of symmetry
Here are examples of questions we've helped users solve.
Graphing and Analyzing Parabolas
Para la función \( f(x) = 2x^2 - 6x + 4 \), necesitamos determinar la orientación, el eje de simetría, el vértice, los ceros (si los hay), las intercepciones con los ejes, el dominio y el recorrido. Primero, encontramos la fórmula del eje de simetría de una parábola de la forma \( ax^2+bx+c \), que es \( x = -\frac{b}{2a} \). <p>\( x = -\frac{-6}{2(2)} = \frac{6}{4} = \frac{3}{2} \)</p> El eje de simetría es: <p>\( x = \frac{3}{2} \)</p> El vértice de la parábola está en el eje de simetría, así que sustituimos \( x \) por \( \frac{3}{2} \) en \( f(x) \) para encontrar el valor de \( y \) en el vértice. <p>\( f(\frac{3}{2}) = 2(\frac{3}{2})^2 - 6(\frac{3}{2}) + 4 \)</p> <p>\( f(\frac{3}{2}) = 2(\frac{9}{4}) - 9 + 4 \)</p> <p>\( f(\frac{3}{2}) = \frac{18}{4} - \frac{36}{4} + \frac{16}{4} \)</p> <p>\( f(\frac{3}{2}) = -\frac{2}{4} = -\frac{1}{2} \)</p> Entonces, el vértice es: <p>\( V(\frac{3}{2}, -\frac{1}{2}) \)</p> Dado que el coeficiente de \( x^2 \) es positivo, la parábola se abre hacia arriba: <p>Orientación: Hacia arriba</p> Para encontrar los ceros, debemos resolver \( 2x^2 - 6x + 4 = 0 \). Podemos utilizar la fórmula cuadrática o factorizar si es posible. La fórmula cuadrática es \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). <p>\( x = \frac{6 \pm \sqrt{(-6)^2 - 4(2)(4)}}{2(2)} \)</p> <p>\( x = \frac{6 \pm \sqrt{36 - 32}}{4} \)</p> <p>\( x = \frac{6 \pm \sqrt{4}}{4} \)</p> <p>\( x = \frac{6 \pm 2}{4} \)</p> Esto nos da dos ceros: <p>\( x = \frac{8}{4} = 2 \) y \( x = \frac{4}{4} = 1 \)</p> Los ceros son: <p>\( x = 1 \) y \( x = 2 \)</p> El dominio de cualquier parábola es todos los números reales: <p>Dominio: \( (-\infty, \infty) \)</p> El recorrido, dado que la parábola se abre hacia arriba y el vértice es el punto más bajo, será de \( y \) en el vértice (mínimo \( y \)) hasta infinito. <p>Recorrido: \( [-\frac{1}{2}, \infty) \)</p> Las intercepciones con los ejes \( x \) son los ceros: <p>\( (1,0) \) y \( (2,0) \)</p> Para el intercepto con el eje \( y \), evaluamos \( f(0) \): <p>\( f(0) = 2(0)^2 - 6(0) + 4 = 4 \)</p> <p>Intercepto en \( y \): \( (0,4) \)</p>
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.
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. \( f(x) = (x + 3)^2 + 3 \).
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.
Identifying a Parabolic Equation with Given Characteristics
The image contains a question asking which equation represents a parabola that opens upward, has a minimum value of 3, and has an axis of symmetry at \( x = -3 \). The options provided are four quadratic equations: A. \( f(x) = -(x + 3)^2 + 3 \) B. \( f(x) = (x - 3)^2 + 6 \) C. \( f(x) = (x + 3)^2 - 6 \) D. \( f(x) = (x + 3)^2 + 3 \) A parabola that opens upward will have the square term with a positive coefficient. Options B, C, and D all have the square term with a positive coefficient, and thus they represent parabolas that open upward. Option A is incorrect because the negative sign in front would make the parabola open downward. Regarding the axis of symmetry, it is determined by the \( h \) in the vertex form of a parabola \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. The axis of symmetry is x = h. So in this case, we need \( h = -3 \). Options C and D both have \( x + 3 \), which can be written as \( x - (-3) \), indicating that the axis of symmetry is at \( x = -3 \), which meets the axis of symmetry criteria. Lastly, the parabola has a minimum value of 3, which means that the vertex is at \( (h, k) = (-3, 3) \). So we want the constant term (after completing the square) to be 3. Only option D has a constant term of +3. The correct answer based on all these conditions is: D. \( f(x) = (x + 3)^2 + 3 \) This equation represents a parabola that opens upward, has its vertex and therefore its minimum value at 3, and has an axis of symmetry at \( x = -3 \).
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
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.
Quadratic Function Characteristics and Equation
The question in the image asks you to consider a quadratic function, \(f\), with the following characteristics: 1. Axis of symmetry is the \(y\)-axis, 2. The \(x\)-intercepts are \( (-4, 0) \) and \( (4, 0) \), 3. The graph cuts the \(y\)-axis at \(y = 16\). Followed by two sub-questions: 1.1. It asks to sketch the graph of \(f\) on a system of axes. Clearly show ALL intersepts with the axes. 1.2. It asks to determine the equation of the graph in the form \(ax^2 + q = y\). Let's address each part: For 1.1, sketching the graph requires understanding those characteristics: - The axis of symmetry being the \(y\)-axis implies that the parabola will be mirrored across the \(y\)-axis. - The \(x\)-intercepts are the points where the graph crosses the \(x\)-axis, which are given as \( (-4, 0) \) and \( (4, 0) \). These are the roots of the quadratic equation. - The graph cuts the \(y\)-axis at \(y = 16\), which means the \(y\)-intercept is at the point \( (0, 16) \). Plotting these points, starting with the \(x\)-intercepts at \(-4\) and \(4\), and the \(y\)-intercept at \(16\), you'll draw a "U"-shaped parabola that opens upwards. For 1.2, to find the equation of the parabola, you can use the vertex form of a quadratic equation, \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola. For this parabola, the vertex is on the \(y\)-axis, so \(h = 0\). Since the axis of symmetry is the \(y\)-axis, the vertex \(h\) is 0. Thus, the equation simplifies to \(y = ax^2 + k\). The \(x\)-intercepts tell us the roots of the quadratic equation are \(x = -4\) and \(x = 4\). Therefore, the factored form of the quadratic is \[ y = a(x + 4)(x - 4). \] Expanding this gives us \[ y = a(x^2 - 16). \] We are also told the \(y\)-intercept is \(y = 16\), so when \(x = 0\): \[ 16 = a(0 - 16), \] which gives \[ 16 = -16a, \] Hence, \[ a = -1. \] So the equation of the parabola is: \[ y = -1(x^2 - 16), \] which simplifies to \[ y = -x^2 + 16. \] The question asks for a form where \(ax^2 + q = y\), so this is \[ -x^2 + 16 = y, \] which, when written in the requested form, is \[ y = -x^2 + 16. \]
Finding the Axis of Symmetry for a Quadratic Equation
La frase "Eje simetría" en la imagen sugiere que se busca determinar el eje de simetría de una parábola. Para una ecuación cuadrática en la forma estándar \( y = ax^2 + bx + c \), el eje de simetría se puede encontrar usando la fórmula \( x = -\frac{b}{2a} \). La ecuación proporcionada en la imagen es: \( -x^2 + 8x - 5 \). Aquí, \( a = -1 \), \( b = 8 \), y \( c = -5 \). Aplicando la fórmula para encontrar el eje de simetría: \( x = -\frac{b}{2a} \) \( x = -\frac{8}{2(-1)} \) \( x = -\frac{8}{-2} \) \( x = 4 \) Por lo tanto, el eje de simetría de la parábola dada por la ecuación \( -x^2 + 8x - 5 \) es \( x = 4 \).
Finding the Axis of Symmetry of a Parabola
Para encontrar el eje de simetría de una parábola representada por una función cuadrática de la forma \( ax^2 + bx + c \), utilizamos la fórmula \( x = -\frac{b}{2a} \). En tu caso, la función cuadrática es \( -x^2 + 8x - 5 \). Aquí, \( a = -1 \) y \( b = 8 \). Sustituimos los valores de \( a \) y \( b \) en la fórmula para el eje de simetría: \( x = -\frac{b}{2a} = -\frac{8}{2 \cdot (-1)} = -\frac{8}{-2} = 4 \). Por lo tanto, el eje de simetría de la parábola dada por la función \( -x^2 + 8x - 5 \) es \( x = 4 \).
Quadratic Equation and Axis of Symmetry
La expresión que tienes en la imagen es una ecuación cuadrática. El "eje simétrico" al que se refiere probablemente es el eje de simetría de la parábola definida por esta ecuación cuadrática. El eje de simetría de una parábola dada por la ecuación cuadrática \( ax^2 + bx + c = 0 \) se encuentra en la línea \( x = -\frac{b}{2a} \). Para la ecuación que has proporcionado \( x^2 + 8x - 5 = 0 \), podemos identificar \( a = 1 \) y \( b = 8 \). Por lo tanto, el eje de simetría es: \[ x = -\frac{8}{2 \cdot 1} = -\frac{8}{2} = -4 \] Así que el eje simétrico de la parábola descrita por la ecuación cuadrática es \( x = -4 \).
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