What is the Axis of Symmetry of a Quadratic Function?Ever notice that the left side of the graph of a quadratic equation looks a lot like the right side of the graph? In fact, these sides are just mirror images of each other! If you were to cut a quadratic equation graph vertically in half at the vertex, you would get these symmetrical sides. That vertical line that you cut has a special name. It's called the axis of symmetry. To learn about the axis of symmetry, watch this tutorial! Show The axis of symmetry of a parabola is a line about which the parabola is symmetrical. When the parabola is vertical, the line of symmetry is vertical. When a quadratic function is graphed in the coordinate plane, the resulting parabola and corresponding axis of symmetry are vertical. STANDARD FORM The graph of the parabola represented by the quadratic function y = a( x - p )2 + q has an axis of symmetry represented by the equation of the vertical line x = p. GENERAL FORM Click here to see how this formula is derived. FACTORED FORM The axis of symmetryAll parabolas have exactly one axis of symmetry (unlike a circle, which has infinitely many axes of symmetry). If the vertex of a parabola is \((k,l)\), then its axis of symmetry has equation \(x=k\). Detailed description of diagram We can find a simple formula for the value of \(k\) in terms of the coefficients of the quadratic. As usual, we complete the square: \begin{align*} y &= ax^2 + bx + c\\ &= a\Big[x^2 + \dfrac{b}{a}x + \dfrac{c}{a}\Big]\\ &= a\Big[\Big(x+\dfrac{b}{2a}\Big)^2 + \dfrac{c}{a} - \Big(\dfrac{b}{2a}\Big)^2\Big]. \end{align*} We can now see that the \(x\)-coordinate of the vertex is \(-\dfrac{b}{2a}\). Thus the equation of the axis of symmetry is \[ x = -\dfrac{b}{2a}. \] We could also find a formula for the \(y\)-coordinate of the vertex, but it is easier simply to substitute the \(x\)-coordinate of the vertex into the original equation \(y=ax^2+bx+c\). ExampleSketch the parabola \(y=2x^2+8x+19\) by finding the vertex and the \(y\)-intercept. Also state the equation of the axis of symmetry. Does the parabola have any \(x\)-intercepts? SolutionHere \(a=2\), \(b=8\) and \(c=19\). So the axis of symmetry has equation \(x=-\dfrac{b}{2a}=-\dfrac{8}{4}=-2\). We substitute \(x=-2\) into the equation to find \(y = 2\times (-2)^2 + 8\times (-2) + 19 = 11\), and so the vertex is at \((-2,11)\). Finally, putting \(x=0\) we see that the \(y\)-intercept is 19. There are no \(x\)-intercepts. Detailed description of diagram Exercise 3A parabola has vertex at \((1,3)\) and passes through the point \((3,11)\). Find its equation. Screencast of exercise 3 Next page - Content - The quadratic formula and the discriminant How do you find the vertex and axis of symmetry?The vertex is the highest point if the parabola opens downward and the lowest point if the parabola opens upward. The axis of symmetry is the line that cuts the parabola into 2 matching halves and the vertex lies on the axis of symmetry.
What is the equation for the axis of symmetry of a parabola?If the vertex of a parabola is (k,l), then its axis of symmetry has equation x=k. We can find a simple formula for the value of k in terms of the coefficients of the quadratic. As usual, we complete the square: y=ax2+bx+c=a[x2+bax+ca]=a[(x+b2a)2+ca−(b2a)2].
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