Graphing Quadratic Equations Quadratic Equation A Quadratic Equation in Standard Form (a, b, and c can have any value, except that a can't be 0.) Here is an example: Quadratic Equation Graphing You can graph a Quadratic Equation using the Function Grapher, but to really understand what is going on, you can make the graph yourself. Read On! The Simplest Quadratic The simplest Quadratic Equation is: f(x) = x2 And its graph is simple too: Square function This is the curve f(x) = x2 It is a parabola. Now let us see what happens when we introduce the "a" value: f(x) = ax2 ax^2 Larger values of a squash the curve inwards Smaller values of a expand it outwards And negative values of a flip it upside down Quadratic Graph Play With It Now is a good time to play with the "Quadratic Equation Explorer" so you can see what different values of a, b and c do. The "General" Quadratic Before graphing we rearrange the equation, from this: f(x) = ax2 + bx + c To this: f(x) = a(x-h)2 + k Where: h = -b/2a k = f( h ) In other words, calculate h (=-b/2a), then find k by calculating the whole equation for x=h First of all ... Why? Well, the wonderful thing about this new form is that h and k show us the very lowest (or very highest) point, called the vertex: And also the curve is symmetrical (mirror image) about the axis that passes through x=h, making it easy to graph quadratic vertex So ... h shows us how far left (or right) the curve has been shifted from x=0 k shows us how far up (or down) the curve has been shifted from y=0 Lets see an example of how to do this: Example: Plot f(x) = 2x2 - 12x + 16 First, let's note down: a = 2, b = -12, and c = 16 Now, what do we know? a is positive, so it is an "upwards" graph ("U" shaped) a is 2, so it is a little "squashed" compared to the x2 graph Next, let's calculate h: h = -b/2a = -(-12)/(2x2) = 3 And next we can calculate k (using h=3): k = f(3) = 2(3)2 - 12·3 + 16 = 18-36+16 = -2 So now we can plot the graph (with real understanding!): 2x^2-12x+16 We also know: the vertex is (3,-2), and the axis is x=3 From A Graph to The Equation What if we have a graph, and want to find an equation? Example: you have just plotted some interesting data, and it looks Quadratic: quadratic data Just knowing those two points we can come up with an equation. Firstly, we know h and k (at the vertex): (h, k) = (1,1) So let's put that into this form of the equation: f(x) = a(x-h)2 + k f(x) = a(x-1)2 + 1 Then we calculate "a": We know (0, 1.5) so: f(0) = 1.5 And we know the function (except for a): f(0) = a(0-1)2 + 1 = 1.5 Simplify: f(0) = a + 1 = 1.5 a = 0.5 And so here is the resulting Quadratic Equation: f(x) = 0.5(x-1)2 + 1 Note: This may not be the correct equation for the data, but it’s a good model and the best we can come up with.