How do you find the range of f(x)=x2+3?

3 Answers
Sep 21, 2015

{yy3}

Explanation:

Since this function is quadratic, its graph is a parabola and it either has a minimum or a maximum value for y. To solve for the minimum/maximum value, we convert our equation to the vertex form y=a(xh)2+k by "completing the square".

The number a determines whether the parabola opens upward or downward. This is important because it will tell us whether we're looking for the minimum or maximum value of y. If it is positive, then we are looking for the minimum value. If it is negative, we are looking for the maximum value. The number then k tells us the minimum/maximum value of y.

Luckily, the function f(x)=x2+3 is already in vertex form. You can look at it this way:
f(x)=(x0)2+3

First, let's look at a. In this equation, a=1. Since it is negative, it means that we are looking for the maximum value of y.

Next, we look at k. In this equation, k=3, meaning 3 is the maximum value for y.

The range will then be {yy3}. You may also write it in set interval notation as (,3].

Sep 21, 2015

(,3]

Explanation:

The given function represents a parabola opening downwards with vertex at (0,3). Hence its range would be (,3]

Sep 21, 2015

I found: <y3

Explanation:

This function is represented graphically by a downwards parabola; this is because the 1 in front of the x2 term.
To find the range (= possible y values) we need to find the highest point reached by our parabola, the vertex.
The x coordinate of the vertex is given as: xv=b2a
where the coefficients banda are found writing the function in general form as:
f(x)=ax2+bx+c=1x2+0x+3
so:
xv=012=0
giving for y (substituting x=0 into your function):
yv=f(0)=3

So the range (= possible y values) is:
<y3

Graphically:
graph{-x^2+3 [-10, 10, -5, 5]}