Identify coefficients

These numbers \( a, b, c\) will be useful for the next few steps.

  • \(a\) is the number before \(x^2\); \(a=1\)

  • \(b\) is the number before \(x\). In our case, \(b =0\) because we do not have a \(bx\) term.

  • \(c\) is the number that does not have any variable attached to it. \(c=4\)

Determine the Vertex + direction of the Parabola

We find the \(x\) coordinate of the vertex by using this formula:
\(x = \frac{-b}{2a}\)

In our case, since \(b =0\), then \(x=0\).

Tip: To save time in other cases, whenever you see \(b=0\), remember that \(x=0\) for the vertex of the parabola.

To find the \(y\) coordinate of the vertex, we substitute this value in the equation and we get:

\(x^2 + 4=0^2+4= 4\)

So the vertex is the point (\(0, 4\)) and the axis of symetry is the y-axis itself. This also means that we have found the y-intercept of this parabola, which is the vertex. Now we just need to add a few points to the left and right of the vertex. Easier than 'regular' cases, right?

Let's add (\(0, 4\)) to the graph.



Direction?

  • Is the parabola going up?

    If \(a\) is a positive number, then the parabola goes up and the vertex is the lowest point in the graph. In our case \(a=1\), which is a positive number, so our parabola goes up.

  • Or down

    If \(a\) is negative, then the parabola will go down and the vertex will be the highest point in the graph.

Add other points

Since we know that the vertex is at (\(0, 4\)) and this parabola goes up (remember that when \(a\) is positive, the parabola goes up), it will not cross the x-axis, so there are no x-intercepts.  We need to find a few points to the left and a few points to the right of the vertex (\(x=0)\). We have to choose an \(x\) smaller than zero, and one bigger than zero.

\(x=-2\) ,  \(y=(-2)^2 + 4 =8\)
So the point (\(-2, 8\)) is part of the parabola, and so is the point (\(2, 8\)) because the y-axis is the simmetry axis.

Now, let's choose another \(x\) that is bigger than zero.
\(x=3\) ,  \(y=3^2 + 4 = 13\)

So the point (\(3, 13\)) is part of the parabola, and so is the point (\(-3, 13\)).

Let's add all these points to the graph. 

Connect the dots

Now that we have enough dots, we can connect them to build our parabola. Remeber that the parabola is a curve, NOT a straight line.