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. Now move the rightmost slider for c and let it settle on, say, 12.Its graph is therefore a horizontal straight line through the origin. This simplifies to y = 0 and is of course zero for all values of x. And we got it right.Since a, b, c are all set to zero, this is the graph of the equation So first I'll do the vertexĪt 2 comma negative 5, which is right there. Which is equal to- let's see, this is equal to 2 squared is 4. When x equals 2, y is going toīe equal to 5 times 2 squared minus 20 times 2 plus 15, To substitute back in to figure out its y-coordinate. Sits exactly smack dab between the roots, I want to figure out, is this point right This is true, and you canĪdd 3 to both sides of this. And so this will be true ifĮither one of these is 0. X's will make this expression 0, and if they make Side, we still have that being equal to 0. On factoring quadratics if this is not so fresh- isĪ negative 3 and negative 1 seem to work. And whose sum is negativeĤ, which tells you well they both must be negative. Whose product is positive 3? The fact that their And now we can attempt toįactor this left-hand side. Plus 15 over 5 is 3 isĮqual to 0 over 5 is just 0. Me- these cancel out and I'm left with x squared The x squared term that's not a 1, is to see if I canĭivide everything by that term to try to simplify I like to do whenever I see a coefficient out here on We're going to try to solve the equation 5x Those three points then I should be all set with And then I also want toįigure out the point exactly in between, which is the vertex. Minus 20x plus 15, when does this equal 0? So I want to figure Seen, intersecting the x-axis is the same thingĪs saying when it does this when does y equal I want to first figure out whereĭoes this parabola intersect the x-axis. You can just take threeĬorresponding values for y are and just graph The following equation y equals 5x squared
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