What is the range of -3x^2+7x-1?

Question

How do I solve this and find the range and domain.

Answer 1

Your first step is to find the vertex, because vertex of a parabola is related to its range.

Note that the coefficient of x^2 is negative, meaning it's a parabola facing down. What you're going to have here is the vertex being the maximum point. Therefore, if your vertex is (h,k), then your range would be (-infinity, k].

To solve for the vertex, you have to use a process called completing the square.

y = -3x^2 + 7x - 1

Your first step would be to factor a -3 out of the first two terms.

y = -3(x^2 - 7/3x) - 1

Now, to complete the square, you will have to take the coefficient of x , and take "half squared" of it. This will be the key number in completing the square. So let's take "half squared" of -7/3.

Half of -7/3 is the same as saying (1/2) times (-7/3), so you get the answer (-7/6). But we want half *squared*, so we square that, and we get 49/36. We have to add this value within the brackets.

y = -3(x^2 - 7/3x) - 1
y = -3(x^2 - 7/3x + 49/36) - 1 + ?

Noticed that we added something new into the equation. We can't add something new without offsetting the value back to the way it was again. For that reason, we have to add a value to offset that.

The value that we added in wasn't really 49/36, but -3 times 49/36, because of the lingering -3 outside of the brackets. Therefore, to offset the function back to the way it was, you have to ADD 3(49/36), or add 49/12

y = -3(x^2 - 7/3x + 49/36) - 1 + 49/12

Now, we have a perfect square in the brackets.

y = -3(x - 7/6)^2 - 1 + 49/12

Let's clean up those whole numbers by merging them into one.

y = -3(x - 7/6)^2 - 12/12 + 49/12
y = -3(x - 7/6)^2 + 37/12

A parabola in standard form is as follows:

y = A(x - h)^2 + k, where (h,k) is the coordinates of the vertex.

In this case, our vertex is clearly located at (7/6, 37/12)

Therefore, our range is (-infinity, 37/12]

Answer 2

f(x) : R(domain) -> { -infinity , 11/3 } (range)
draw the graph of tis function to find out its range