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The axis of symmetry A parabola is symmetrical about its axis of symmetry. axis of symmetry axis of symmetry vertex vertex The vertex is the turning point where the axis of symmetry cuts the parabola. • For a parabola that is concave up, the vertex is a minimum turning point. • For a parabola that is concave down, the vertex is a maximum turning point. Any horizontal line drawn across the parabola will intersect the parabola at equal distances on either side of the axis of symmetry. So you only really need to draw half of the parabola on one side of its axis, and then reflect this across the axis to complete the parabola. You can locate the axis of symmetry of a parabola by drawing a horizontal interval across the parabola. The axis runs vertically through the midpoint of that interval. Part 1 Parabolas 1 Follow through the steps in this example. Do your own working in the margin if you wish. a What is the equation of the axis of symmetry of the parabola y = x 2 4x + 8 shown? (0, 8) b (4, 8) What are the coordinates of its vertex? Solution a The axis runs through the midpoint of (0, 8) and (4, 8). You can easily see this point to be (2, 8). (You don’t need to use the midpoint formula for this.) The equation for the axis of symmetry is x = 2 . b The vertex has x-value of 2. Substituting this into the equation of the parabola, y = x 2 4x + 8 y = (2)2 4(2) + 8 = 48+8 =4 The vertex is (2, 4). 2 PAS5.3.4 Coordinate geometry Activity – The axis of symmetry Try these. 1 a Show that the points (1, 4) and ( 2 12 , 4) lie on the parabola y = 2x 2 3x + 9 . ___________________________________________________ ___________________________________________________ ___________________________________________________ ___________________________________________________ ___________________________________________________ b What is the equation of the axis of symmetry? ___________________________________________________ ___________________________________________________ ___________________________________________________ c What are the coordinates of the vertex for this parabola? ___________________________________________________ ___________________________________________________ ___________________________________________________ d Find the x- and y-intercepts for this parabola. ___________________________________________________ ___________________________________________________ ___________________________________________________ e Part 1 Parabolas Sketch this parabola showing the above features. 3 Activity – The axis of symmetry 1 Substitute both these points into the equation of the parabola, showing each time the result is true. a y = 2x 2 3x + 9 4 = 2(1)2 3(1) + 9 4 = 2 3 + 9 4=4 As this is true, (1, 4) lies on the parabola. y = 2x 2 3x + 9 4 = 2(2 12 )2 3(2 12 ) + 9 4 = 12 12 + 7 12 + 9 4=4 1 As this is true, (2 ,4) lies on the parabola. 2 b The axis of symmetry lies at the midpoint of (1, 4) and 1 3 (2 ,4) , which is ( ,4) . The equation of the axis of 2 4 3 symmetry is x = . 4 c To find the vertex, substitute x = 3 into y = 2x 2 3x + 9 . 4 y = 2x 2 3x + 9 3 3 = 2 3 + 9 4 4 1 = 10 8 3 1 The vertex is ,10 4 8 2 4 PAS5.3.4 Coordinate geometry d For the x-intercept, y = 0. y = 2x 2 3x + 9 = 2(0)2 3(0) + 9 =9 The curve crosses the y-axis at y = 9 . (Or simply look for the constant term.) For the y-intercept, x = 0 . y = 2x 2 3x + 9 2x 2 3x + 9 = 0 (x + 3)(3 2x) = 0 x + 3 = 0or3 2x = 0 1 x = 3x = 1 2 The curve crosses the x-axis at –3 and 1.5. y e 10 9 8 7 6 5 4 3 2 1 –3 –2 –1 0 –1 1 2 x –2 Part 1 Parabolas 5