Question

How do you find the domain and range of `y =2x^2 - x - 6`?

Answer 1

Domain: All Real Numbers
Range: -6`<=`y`<`∞

Explanation:

This equation is a quadratic equation with a parabola as its graph. Its domain consists of all real numbers because its graph extends infinitely in the negative and positive directions (left and right) on the x-axis.

The range is not as simple, however.
A parabola will always extend infinitely in either the negative or positive direction (down or up) on the y-axis, not both.

To determine the vertical direction in which the graph extends INFINITELY, look at the coefficient of the `x^2` term in your equation. If the number is positive, your graph extends infinitely in the positive vertical direction (up). If it is negative, it extends infinitely in the negative vertical direction (down).

Look at the "`x^2`" term in your equation (`2x^2`). The coefficient here (2) is positive. Therefore, your equation extends infinitely upwards. This is only half of your final answer (∞).

Now we must find the y-value of the lowest point on this graph in order to determine the second half of your answer. This lowest point is known as "the vertex" of your graph. Fortunately, you may use your equation to find out what this lowest point is.

The standard form of a quadratic equation is:
`Ax^2+Bx+C`

Compare this with your equation, `2x^2-x-6`. They look similar, right?

If we set these two equations equal to one another, we find that
A = 2, B = -1, and C = -6.

We can use these values to help find your graph's vertex. The vertex lies on the axis of symmetry, which can be found with the equation:
x = -B/2A

If we plug in our values, we get:
x = -(-1)/2*2 = 1/4 (this is the x-value of your vertex)

Plug this answer back into your equation to find the y-axis of your vertex: y = `2(1/4)^2-(1/4)-6` = -6. This answer is the second half of your final answer.

Remember that your range starts from the the lowest y-coordinate (-6) to the highest y-coordinate (∞) of your graph.

Your range is -6`<=`y`<`∞

Keep in mind that x or y can never be `<=` ∞ or `>=` -∞ because infinite is not a definite number that a variable such as x or y can equate to. Therefore, NEVER say that x or y `<=` ∞ or `>=` -∞.