Finding the roots, or values where a graph crosses the x-axis, is a common problem that arises in math. Fortunately, there are some simple rules you can follow to find these roots!
Whenever you are solving for where a graph crosses the x-axis, you are actually solving for what value(s) of m (where m represents the x-coordinate) the equation y = mx2 – 5x – 2 equals zero.
This equation is the formula for a parabola, so you can also think of this solution in terms of how a parabola functions. Parabolas have two defining characteristics: they curve upwards and their axis of symmetry is parallel to the x-axis.
By looking at these properties, you can determine whether or not a graph has any X-intercepts (where the graph crosses the x-axis). If the axis of symmetry is parallel to the x-axis, then it cannot cross it at any point.
X = -2 is a X-intercept
A x-intercept is a value where the graph crosses the x-axis. In other words, it is a value where y = 0.
When the equation for the line is Y = Mx2 – 5x – 2, then M is any real number and not -2, so -2 cannot be part of the equation. -2 is not a valid solution to the equation.
Therefore, when M = -2, there is no point where y = 0. This means that there is no x-intercept at -2, which is incorrect.
For any other value of M, there are x-intercepts at some values of x. For example, if M = 2 then there are x-intercepts at -1 and 2. These are points where y = 0.
X = 2 is a X-intercept
A x-intercept is a value of x for which the equation Y = Mx2 – 5x – 2 has a y-value of 0. In other words, when the variable Y equals zero, then the variable X must equal 2, because that is the only value of X that makes the equation true.
The graph does not have any x-intercepts when M = 0. When M = 0, the graph has no y-axis intercepts because there is no y-value of zero when x = 0. There are no x-intercepts because there are no values of X where Y equals zero.
For any positive value of M, there are at least two distinct values of X for which Y equals zero. For these values of M, there will be x-intercepts.
Y = Mx2 – 5x – 2 has no x-intercepts for M > 4
In this section, we will discuss when the graph of Y = Mx2 – 5x – 2 has no x-intercepts. When the value of M (which stands for magnitude) is greater than 4, the graph of Y = Mx2 – 5x – 2 has no x-intercepts.
We can prove this by first noticing that the equation of the graph of Y = Mx2 – 5x – 2 is y = (M–5)x2 − 2. Now, observe that the coefficient of x2 is negative, and that it is greater than 1. Therefore, y>0 when x 4.
You can also observe that as M increases, so does the magnitude or length of the curve created by Y=MX2−5X−2. Therefore, as the curve becomes longer and wider, there are fewer points where it intersects with the horizontal axis.
Proof for no x-intercepts for M > 4
To prove that the graph of Y = Mx2 – 5x – 2 has no x-intercepts for M > 4, we must find a case where the graph does have an x-intercept and show how this can be proven false.
To do this, we will take the derivative of the equation and solve for x. We will then plot this equation to see if it crosses the x-axis.
y =mx2 – 5x – 2 = 0 + (-2) + (5*2) – (–5)*(–2) = 0 + 10 + 10 = 20
Thus, there is an implicit solution of (0,20), which means that there is an x-intercept at (0,20). To prove that this point is not on the graph, we must simply plot both and see that they are different points.
Calculate y when x = -1
In this case, y = 2 when x = -1, so the graph of Y = Mx2 – 5x – 2 has a vertical asymptote at the value of 2 for all values of x.
As x increases in value, the graph gets closer and closer to the value of 2, but never touches it. This means that there are no intercepts for y = 2.
For any given value of M, if the equation Y = Mx2 – 5x – 2 has no X-intercepts, then there is a constant value of y such that y = 2. This is very interesting!
This fact can be generalized: For what values of M does the graph of Y = Mx2 – 5x – 2 have no X-intercepts? The answer is: When M is not equal to 1 or -1.
Find the derivative of y = mx² − 5x − 2
The next step is to find the derivative of y = mx² − 5x − 2. The derivative is the equation that describes the rate of change of y = mx² − 5x − 2, which in this case is change in y over change in x.
The derivative is written as: dY/dx=mx²−5x−2. This means that as x increases by one unit, Y increases by m²−5x−2 units.
As mentioned before, if there are no x-intercepts, then there cannot be any 0s in the derivative. So, let’s see if we can factor out a 0 from m²−5x−2 to see if it plays any role. We can’t, so it does not matter whether it is present or not.
Set the derivative to 0 and solve for m
So, let’s go back to our original question: For what values of m does the graph of y = mx2 – 5x – 2 have no x-intercepts?
We need to figure out where the graph has no x-intercepts, which means we need to figure out where the graph is rising or descending. To do this, we need to look at the curve of the graph, which means we need to look at the values of m.
If m is greater than 1, then the equation is increasing, so there cannot be any x-intercepts. If m is equal to 1, then there is a single point of inflection, which means there cannot be an x-intercept at that point. If m is less than 1, then the equation is decreasing, so there cannot be any x-intercepts.
Check your solution by calculating the y-value when x = 1 and x = 2
A quick way to check your solution is to calculate the y-value when x = 1 and x = 2. If the y-values you get are different, then you know that there is at least one x-intercept.
By checking these two values, you can determine whether or not the equation has x-intercepts or not. For example, if when x = 1, y = -2, and when x = 2, y = 0, then there is no solution for m where there are no x-intercepts.
There are many ways to solve this equation. You do not have to use elimination if it is not helpful.
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