SOLUTION: I have three quadratic equations that need to be solved and graphed on one graph to show the relationship between the size of three curves. we choose at least 6 plots

SOLUTION: I have three quadratic equations that need to be solved and graphed on one graph to show the relationship between the size of three curves. we choose at least 6 plots - I chose

Question 138950: I have three quadratic equations that need to be solved and graphed on one graph to show the relationship between the size of three curves.
we choose at least 6 plots - I chose for X - o,1,2,3,-1,-2,-3. I have to solve for Y.
The equations are:
1. y=2x^+10x+12
2. y= -1/2x^+4x+3
3. 2y=x^+4x+5
My problem has been how to solve these type of equations
Any help would be appreciated
Thank YOu
Sorry, my name is Leonora
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
First, put number 3 in standard form by dividing by the coefficient on y. . Now it is in form.

Now, instead of just choosing values and trying to calculate the value of the function for each, you need to find the critical points and characteristics.

First thing to notice is that the sign on the lead coefficient tells you which way the curve opens, up or down. Positive opens up, negative opens down. So 1 and 3 open upward and 2 opens down.

Next thing to find is the axis of symmetry. This is a vertical line about which the curve is symmetrical -- the two parts of the graph on either side of the axis of symmetry are mirror images of each other. Once you have your equation in standard form, this is a simple calculation. Take the negative of the coefficient on the 1st degree term (the 'b' in the standard form) and divide by 2 times the lead coefficient (the 'a'). So calculate . For your first equation, you should get . The axis of symmetry is then the vertical line: . for your first equation, for example.

The first point you want to find is the vertex of your parabola. That is the point at the very bottom (or top for #2) of the curve. You already have the x-coordinate of this point -- it is the same as the value you calculated for the axis of symmetry. The y-coordinate of this point is found by substituting the x-coordinate value into your function and calculating the result.

The next points to find are the x-intercepts. This is where the curve intersects the x-axis. Since the y-coordinate of any point on the x-axis is zero, just set the function equal to zero and solve the quadratic equation. You can factor it or use the quadratic formula. If you got a positive value for the y-coordinate when you calculated the vertex for a 'opens upward' curve, or you got a negative value when you did it for the 'opens downward' curve, then you can skip this step because the curve doesn't intersect the axis. If you got a zero result when you calculated the y-coordinate of the vertex, then you can also skip this step because the vertex and the intersection of the curve with the x-axis are the same place.

You can also find the y-intercept. This is the one place where the curve intersects the y-axis. Points on the y-axis have x-coordinates that are all 0, so just substitute 0 for x in your function and solve (you just get the value of the constant term). Since the curve is symmetrical, take the value of the x-coordinate of the vertex and double it, then you will have a point at this new x-coordinate and the value of the y-intercept.

If you think you need a couple more points to get a smooth curve, go to the nearest whole unit one side or the other of the vertex, then calculate the value and plot your point. Because of symmetry, you will have a point with the same y value the same distance on the other side of the vertex.

Let's do the first one. We have already established that the axis of symmetry is , so the x-coordinate of the vertex is . The y-coordinate of the vertex is then . So, the vertex is at (,)

This is an "opens upward" with a negative y-coordinate on the vertex, meaning that the curve will intersect the x-axis in two points, so let's solve the quadratic to find them:

So the function has zeros at -2 and -3, and the x-intercepts are at (-2,0) and (-3,0)

The y-intercept: Substitute 0 for x: , so the y-intercept is (0,12). The mirror image point is at 2 times the x-coordinate of the vertex, or , so the point is (-5,12).

The next whole number from the x-coordinate of the vertex is -2 or -3, but we already have those points as x-intercepts, so lets pick -1. , so we have a point at (-1,4). By symmetry, we also have a point at (-4,4).

Here's a picture of what we have so far (the green line is the axis of symmetry):

Now, go through this whole process again with the other two functions.
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