Graphing a Quadratic Function

Q:  Graph the Quadratic Function:  f(x) = 2x2 – 3x + 1

Answer:

There are two main methods to do this.  I will do both methods (depending on what you have learned in class, you will want to pick either method 1 or method 2 that I show).

Method 1:  Plotting Points

Because this is a quadratic equation, we know the graph will be in the shape of a parabola.  So, I am going to pick a few values for x, then find the y values.  Then, I will plot the points on the graph to make a parabola.  You can pick any values for x when you do this method.

Let’s pick x = 0.  So, plug x into the equation to find y:

2(0)2 – 3(0) + 1 = 0 – 0 + 1 = 1

Point 1:  When x = 0, y = 1:  (0, 1)


Let’s pick x = 1 now:

2(1)2 – 3(1) + 1= 2(1) – 3 + 1 = 2 – 3 + 1 = 0

Point 2:  When x = 1, y = 0:  (1, 0)


Let’s pick x = -1 now:

2(-1)2 – 3(-1) + 1= 2(1) + 3 + 1 = 2 + 3 + 1 = 6

Point 3:  When x = -1, y = 6:  (-1, 6)

Now, plot the points (0, 1) and (1, 0) and (-1, 6) on a graph.  Connect the dots to make a parabola!

Keep in mind:  You can get more than 3 points for more accuracy.

Method 2:  Finding the y-intercept and the vertex:

The y-intercept of any type of graph occurs when you plug in 0 for x.  So, plug in 0 for x into the equation and solve for y:

2(0)2 – 3(0) + 1 = 0 – 0 + 1 = 1

So, the y-intercept is the point (0, 1).

Now, there is a formula to find the vertex of a quadratic equation.  It goes as follows:

Step 1 to find the vertex:  x = -b / 2a (this will give us the x coordinate of the vertex.

Recall, in our equation, a = 2, b = -3 and c = 1

So, the x part of the vertex is x = -(-3) / 2(2) = 3/4

Now, to get the y part, you need to plug the x part into the equation and solve:

2(3/4)2 – 3(3/4) + 1 = -1/8


The vertex is the point (3/4 , -1/8)

So, on your graph, plot the vertex and the y-intercept.  Draw a parabola through the y-intercept until you hit the vertex.  You know the vertex is either the high or the low point.  Then, “bounce off” the vertex and continue drawing the other half of the parabola.