The roots, or values where the function equals zero, of a function are an important part of understanding the function. How the roots are found can tell you a lot about the function itself.
For example, if finding the root of a function was easy, then it would not be very useful information. This is because it would mean that any number could be substituted for x in the equation and you would get the same answer back.
It takes some advanced math to learn how to find roots of functions that are not simple fractions or integers. Today, we are going to look at some tips on how to identify the roots of a very common type of graph: quadratics.
Quadratics are graphs that have four points where the graph equals zero. These points are called intercepts, and identifying them is key to finding the roots of a quadratic graph.
The roots of this parabola
A parabola is a graph of a function. The function is the relationship between the variables x and y, where x is the independent variable and y is the dependent variable.
Parabolas can be shifted, translated, and rotated, but only in ways that do not change their shape. When doing these transformations, you can find the roots of the parabola by using algebra.
The roots of a parabola are where the graph meets the x-axis. These points are called intercepts because they intercept the graph of the function with the x-axis. There are two roots for every parabola, one on each side of the axis.
In this case, we will look at how to find one of the roots of this particular parabola.
The function is not defined at these roots
While the graph shows roots of the function, the function itself is not defined at these roots. This is clear in that there is no value for the function at these roots.
Theoretically, one could say that the graph shows a continuous curve with two sharp spikes, but this would be very abstract. It would also be difficult to determine where exactly the root is since there would be no single value for the function at that point.
The graph clearly shows how important it is to consider all possible values of a variable when evaluating a function. If only one variable were considered, then there may be some cases where the function is not defined and this may not be noticed.
In this case, checking only whether or not the function is defined at -1 or 1 would be sufficient, and all other cases could be ignored.
Plugging in values for X
The last step in solving a linear equation is to check your answer by plugging it into the original equation. This is known as solving by substitution.
To do this, you need to pick a number for the variable that you are solving for, in this case X. You then solve the linear equation for X using the methods we learned before and then plug in your number for X. If the result is what you wanted, then you solved the linear equation!
Solving by substitution is very important because it can be hard to tell if you solved the linear equation correctly if you do not check it against the original equation. Check out this article to learn more about how to do this!
The last function we looked at was F(x) = x2 – 2x – 3. To solve this, we had to find x such that x2 – 2x – 3 = 0. (Brief refresher: zero means equal to 0.
Determining the y-intercept
The y-intercept of a function is the value where the graph of the function intersects the y-axis.
To find the y-intercept, you need to find where the function equals zero. Zero is the bottommost value of the y-axis, so finding where the function equals zero means finding where it declines to zero.
In this case, we are given that F(x) = X2 – 2x – 3, so we can replace x with –1 to find where it declines to zero. The graph will then intersect the y-axis at –1 on the x-axis.
The blue dot in the image above shows where this intersection occurs. The red dot shows where x = 0, but this does not match up with any point on our graph.
Determining the vertex
The vertex, or bottom, of the parabola is where the curve changes direction. The x-coordinate of the vertex is –1, so F(–1) = (–1)2 – 2(–1) – 3 = 2 – 2 – 3 = 1.
The y-coordinate of the vertex is 1, so F(1) = 12 – 2(1) – 3 = 1 + 0 + 3 = 4.
For example, try substituting –1 for X: f(-1) = (-1)2 − 2(-1) − 3= 1−2−3=0.
Solving for X using algebraic methods
Solving for the variable X in the function f(x) = x2 – 2x – 3 using algebraic methods is possible, although it is more complicated than changing the X to a number.
First, you would have to find two solutions for x2 – 2x – 3 using algebra. Then, you would have to find two values of x that solve the equation x2 – 2x – 3 = 0. The last step is to combine these solutions into one solution for X.
For example, let’s take a look at the roots of the function f(x) = x2–2x–3:
f(x) = (x−1)(x+1)
So there are two possible values of X: 1 or -1.
But because we know that -1 does not satisfy the equation, we can eliminate it as a solution.
Using a graphing calculator or computer program
A graphing calculator or computer program can be used to find the roots of a function. Many calculators have a special feature that allows you to find the x-intercepts, or values where the graph crosses the x-axis, of a function.
This is done by entering the equation of the function into the calculator and then pressing a button that says “find x-intercepts” or something similar.
A computer programming language called C++ was used to create this feature on most calculators, but some use different languages. Check your calculator’s user guide to see what language your calculator uses and how to use it.
Once the x-intercepts are found, you can draw a line through them to see where the roots are located.
The roots of a parabola determine where it crosses the x-axis and where it touches the y-axis
A parabola is a U-shaped curve. It has a value for its height and a value for its width, and it crosses the x-axis when its height is zero.
Parabolas can be shifted left or right along the x-axis, or turned upside down, and they will still be parabolas. They can also be mirrored across the y-axis, making them look like mirrors of each other. Again, they will still be parabolas.
The roots of a parabola are where it crosses the x-axis. There can be one root (a point where the curve touches the x-axis), two roots (where it touches twice), or no roots (where it goes straight down to the x-axis).
The following graph shows all four possibilities for roots of a parabola.
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