How to Find the Range of a Function in Math
Write down the formula., Find the vertex of the function if it's quadratic., Find a few other points in the function., Find the range on the graph.
Step-by-Step Guide
-
Step 1: Write down the formula.
Let's say the formula you're working with is the following: f(x) = 3x2 + 6x
-2.
This means that when you place any x into the equation, you'll get your y value.
This is the function of a parabola. -
Step 2: Find the vertex of the function if it's quadratic.
If you're working with a straight line or any function with a polynomial of an odd number, such as f(x) = 6x3+2x + 7, you can skip this step.
But if you're working with a parabola, or any equation where the x-coordinate is squared or raised to an even power, you'll need to plot the vertex.
To do this, just use the formula
-b/2a to get the x coordinate of the function 3x2 + 6x
-2, where 3 = a, 6 = b, and
-2 = c.
In this case
-b is
-6, and 2a is 6, so the x-coordinate is
-6/6, or
-1.Now, plug
-1 into the function to get the y-coordinate. f(-1) = 3(-1)2 + 6(-1)
-2 = 3
- 6
-2 =
-5.
The vertex is (-1,-5).
Graph it by drawing a point where the x coordinate is
-1 and where the y-coordinate is
-5.
It should be in the third quadrant of the graph. , To get a sense of the function, you should plug in a few other x-coordinates so you can get a sense of what the function looks like before you start to look for the range.
Since it's a parabola and the x2 coordinate is positive, it'll be pointing upward.
But just to cover your bases, let's plug in some x-coordinates to see what y coordinates they yield: f(-2) = 3(-2)2 + 6(-2)
-2 =
-2.
One point on the graph is (-2,
-2) f(0) = 3(0)2 + 6(0)
-2 =
-2.
Another point on the graph is (0,-2) f(1) = 3(1)2 + 6(1)
-2 =
7.
A third point on the graph is (1, 7). , Now, look at the y-coordinates on the graph and find the lowest point at which the graph touches a y-coordinate.
In this case, the lowest y-coordinate is at the vertex,
-5, and the graph extends infinitely above this point.
This means that the range of the function is y = all real numbers ≥
-5. -
Step 3: Find a few other points in the function.
-
Step 4: Find the range on the graph.
Detailed Guide
Let's say the formula you're working with is the following: f(x) = 3x2 + 6x
-2.
This means that when you place any x into the equation, you'll get your y value.
This is the function of a parabola.
If you're working with a straight line or any function with a polynomial of an odd number, such as f(x) = 6x3+2x + 7, you can skip this step.
But if you're working with a parabola, or any equation where the x-coordinate is squared or raised to an even power, you'll need to plot the vertex.
To do this, just use the formula
-b/2a to get the x coordinate of the function 3x2 + 6x
-2, where 3 = a, 6 = b, and
-2 = c.
In this case
-b is
-6, and 2a is 6, so the x-coordinate is
-6/6, or
-1.Now, plug
-1 into the function to get the y-coordinate. f(-1) = 3(-1)2 + 6(-1)
-2 = 3
- 6
-2 =
-5.
The vertex is (-1,-5).
Graph it by drawing a point where the x coordinate is
-1 and where the y-coordinate is
-5.
It should be in the third quadrant of the graph. , To get a sense of the function, you should plug in a few other x-coordinates so you can get a sense of what the function looks like before you start to look for the range.
Since it's a parabola and the x2 coordinate is positive, it'll be pointing upward.
But just to cover your bases, let's plug in some x-coordinates to see what y coordinates they yield: f(-2) = 3(-2)2 + 6(-2)
-2 =
-2.
One point on the graph is (-2,
-2) f(0) = 3(0)2 + 6(0)
-2 =
-2.
Another point on the graph is (0,-2) f(1) = 3(1)2 + 6(1)
-2 =
7.
A third point on the graph is (1, 7). , Now, look at the y-coordinates on the graph and find the lowest point at which the graph touches a y-coordinate.
In this case, the lowest y-coordinate is at the vertex,
-5, and the graph extends infinitely above this point.
This means that the range of the function is y = all real numbers ≥
-5.
About the Author
Kyle Torres
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