First solve these equations algebraically [math]y=-x^2+4x+2[/math] and [math]y=x+2[/math] . [br]After find the solutions for x and y, check show function to show the graphs, then check show intersection to show the (x,y). In others the solution of simultaneous equations is / are the points of intersections of graphs in this case a parabola and a straight line.
Change the value 'm' and 'c' to 0. Observe what are the intersection points. Jot down the values. Do you find anything interseting ? or what these values are called with respect to a function or an equation?
The solutions are as same roots of the equation [math]-x^2+4x+2=0[/math]. Thus we can also that x-intercepts of graph f(x) are the points of intersection of f(x) and straight line y = 0
Let us continue with 'm'= 0 and change 'c'. Thus the straight line will be parallel to ? [br]quadratic equation continue to be [img]data:image/png;base64,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[/img][br]move 'c' to -1, -2 , - 3. jot down straight line equation and the corresponding points of intersection. [br]you can further go up i.e. by moving 'c' to positive values. Again jot down the points of intersection.
Continue with 'm' = 0 and play around with 'c' only. For what values of 'c' such the x-values are always positive? What happens when c = 6?
If you answer as c= 3, 4 and 5, can you generalise? [br]When c = 6 there is only one point.
What happens when c= 7 ? How can you generalise?
There are no points of intersection. We can definitely say for [math]-x^2+4x+2=c[/math][br]1) have no points of intersection when c > 6[br]2) have only one of point when c = 6[br]3) have only positive roots when c is between 2 and 6 [br]4) have negative and positive roots when c is less than 2
For what values 'c' of [math]f\left(x\right)=x^2-4x-1[/math] and [math]g\left(x\right)=x+c[/math] such that f(x) and g(x) do not intersect in a real Cartesian plane?
equate f(x) = g(x) . Using the nature of roots we say that there does not exist real values.