This graph and worksheet allows you to investigate how the constants a, h and k affect the appearance of a parabola whose equation is written in Vertex form.
It may take a minute for the GeoGebra applet below to load, so please be patient (and give it permission to load if you receive any warnings from your browser)
Please scroll down and read through the text below the GeoGebra applet for some suggested tasks that will help you understand how this parabola's appearance is affected by each of the coefficients:
y = a(x - h)^2 + k
If you have comments or suggestions for this page, please click here to make them via my blog entry relating to these applets.
Drag the points a, h, and k along the green sliders at the bottom center of the graph and see what happens. Can you:
- Make the curve go through the origin?
- Make the curve go through the point (-3,-1)?
- Make the curve have a wider opening?
- Make the curve have a narrower opening?
- Make the curve open sideways?
- Make the curve open down instead of up?
- Make the curve into a line?
Which of the above could NOT be achieved? Why not?
Vertex form is most easily understood as a translation and dilation of the most basic parent function possible for a parabola:
y = x ^ 2
This parabola opens up and has its vertex at the origin.
Translating this horizontally by h produces
y = (x - h) ^ 2
This parabola will still open up, but the vertex (and thus the entire parabola) has been shifted horizontally by h so that the vertex still lies on the x-axis, but is no longer at the origin (unless h is zero).
Taking this result and translating it vertically by k produces
y = (x - h) ^ 2 + k
This parabola still opens up, and has still be translated horizontally by h, but has also been translated (vertex and all) vertically by k. If k is not zero, then the vertex no longer lies on the x-axis.
And finally, adding a dilation/scaling factor a that allows us to control the direction (opening up or down) as well as how quickly or slowly it grows produces
y = a(x - h) ^ 2 + k
Note that it can be helpful to subtract k from both sides of an equation in Vertex Form:
y = a(x - h) ^ 2 + k
(y - k) = a(x - h) ^ 2
In this form, you can analyze both x and y translations in exactly the same manner: what value of the variable (x or y) will make the expression in parentheses equal to zero? THAT is the size and direction of the translation along that variable's axis.
Do you remember how to find the y-intercept of a function, the one above in particular? Recall, the y-intercept must lie on the y-axis, so what must its x-coordinate be? Plug this x-value into the equation above, and simplify to find the y-intercept.
Do you remember how to find the x-intercept(s), if any, of a quadratic function? They must lie on the x-axis, so what must their y-coordinate(s) be? Plug this y-value into the equation above, and solve for x to find the x-intercepts. Note that this is fairly easy to do with Vertex Form... why is this so much easier to do with Vertex Form than the Standard Form of the same equation?
Can you convert the last equation above into Standard Form? Carry out the square, then distribute the a and collect like terms (remember that both h and k are constants). Try solving this form for x! Not as easy is it?
What is the name of the procedure one follows to convert Standard Form into Vertex Form? It reverses the process you just went through above, and also allows you to derive the Quadratic Formula if you start from the generic quadratic trinomial and solve for the x-intercepts:
0 = ax^2 + bx + c