![]() ![]() Note thatif a = 0, the x 2 term would disappear and we wouldhave a linear equation! Thus, the standardized form of a quadratic equation is ax 2+ bx + c = 0, where "a" does not equal 0. Simply, the three terms include one that hasan x 2, one has an x, and one term is "by itself"with no x 2 or x. Normally, we see thestandard quadratic equation written as the sum of three termsset equal to zero. So, for our purposes, we willbe working with quadratic equations which mean that the highestdegree we'll be encountering is a square. In an algebraic sense, the definition ofsomething quadratic involves the square and no higher power ofan unknown quantity second degree. Similarly, one of the definitions of the termquadratic is a square. In the figure above, set a to zero and moving the other sliders, convince yourself there can be no axis of symmetry with a = 0.The term quadratic comes from the word quadrate meaning squareor rectangular. If a is zero, there is no axis of symmetry and this formula will not work, the attempt to divide by zero will give an undefined result. When the quadratic is in normal form, as it is here, we can find the axis of symmetry from the formula below. Note too that the roots are equally spaced on each side of it. (We say the curve is symmetrical about this line). Note how the curve is a mirror image on the left and right of the line. This is a vertical line through the vertex of the curve. If the expression inside the square root is negative, the curve does not intersect the x-axis and there are no real roots.Ĭlick on "show axis of symmetry". It gives the location on the x-axis of the two roots and will only work if a is non-zero. ![]() When expressed in normal form, the roots of the quadratic are given by the formula below. Notice that if b = 0, then the roots are evenly spaced on each side of the origin, for example +2 and -2. Under some conditions the curve never intersects the x-axis and so the equation has no real roots. If you make b and c zero, you will see that both roots are in the same place. If the curve does not intersect the x-axis at all, the quadratic has no real roots. Under some circumstances the two roots may have the same value. There are two roots since the curve intersects the x-axis twice, so there are two different values of x where y = 0. In the figure above, click on 'show roots'.Īs you play with the quadratic, note that the roots are where the curve intersects the x axis, where Changing a alters the curvature of the parabolic element.Note how it combines the effects of the three terms. This is the graph of the equation y = 2x 2+3x+4. Note also the roots of the equation (where y is zero) are at the origin and so are both zero. When a is negative it slopes downwards each side of the origin. This is the graph of the equation y = 3x 2+0x+0.Įquations of this form and are in the shape of a parabola, and sinceĪ is positive, it goes upwards on each side of the origin.Īs a gets larger the parabola gets steeper and 'narrower'. Move the left slider to get different values of a.To get a feel for the effects of their values on the graph. This is the equation of y = bx+c and combines the effects of the Now move both sliders b and c to some value.Since the slope is positive, the line slopes up and to the right.Īdjust the b slider and observe the results, including negative values. That is, y increases by 2 every time x increases by one. This is a simple linear equation and so is a straight line whose slope is 2. This is the graph of the equation y = 0x 2+2x+0 which simplifies to y = 2x. Move the center slider to get different values of b.Play with different values of c and observe the result. It is therefore a straight horizontal line through 12 on the y axis. This simplifies to y = 12 and so the function has the value 12 for all values of x. ![]() This is the graph of the equation y = 0x 2+0x+12.
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