Question

Graph the functions f ( x ) = x^2 , g ( x ) = ( x + 4 )^2 , and h ( x ) = ( x − 4 )^2 on the same rectangular coordinate system. Then describe what effect adding a constant, h , to the function has on the vertex of the basic parabola.

Answer 1

In order to graph each parabola, first let's identify the vertex of each one.

To do so, let's compare them with the vertex form of the quadratic equation:

`y=a(x-h)^2+k`

Where the vertex is located at (h, k).

For f(x), we have h = 0 and k = 0, so the vertex is at (0, 0).

For g(x), we have h = -4 and k = 0, so the vertex is at (-4, 0).

For h(x), we have h = 4 and k = 0, so the vertex is at (4, 0).

Now, to graph these functions, we need another point for each one.

Let's use x = 1 in each function and calculate the value of y:

`\begin{gathered} f(1)=1^2=1\\ \\ g(1)=(1+4)^2=25\\ \\ h(1)=(1-4)^2=9 \end{gathered}`

Graphing each parabola using the vertex and the additional point, we have:

(red = f(x), blue = g(x), green = h(x))

The effect of adding a positive constant h (if we have p(x) = (x + h)²) is moving the vertex left by h units.

The effect of adding a positive constant h (if we have p(x) = (x - h)²) is moving the vertex right by h units.