In this video, I explain how to graph a parabola.
The typical equation of a parabola is y = ax2 + bx + c. From the values of a, b and c, we work out up to four different points on the parabola.
First of all, there are the intercepts – where the curve of the parabola (a nice bowl shape) meets the x and y axes.
The y intercept is very easy to find. Put x=0, and you get y=c.
The x intercepts might be a bit harder to find. You put y=0, and then solve the equation ax2 + bx + c = 0. You do this either by factoring the equation, using the quadratic formula, or via some other method such as completing the square.
Finally, you find the vertex of the parabola. This is the bottom (or top) of the bowl shape. It also gives the x coordinate of the axis of symmetry of the parabola. The x-coordinate of the parabola is given by x=-b/(2a). You find the y coordinate by taking this x value and putting it into the equation of the parabola.
For example, with the parabola 2x2 – 3x – 2, you have a=2, b=-3 and c=-2. The y intercept is (0,-2). To find the x intercept of 2x2-3x-2, we need to solve 2x2-3x-2=0. If we factorise the quadratic equation, we get (2x+1)(x-2)=0, so x=-1/2 or x=2. This gives two more points on the parabola, (-1/2,0) and (2,0).
To find the vertex, we use x=-(-3)/(2 x 2) = 3/4. Then, we get y=-3.125. This gives the fourth and last point on the parabola, (3/4,-3.125).
After that, we plot these four points on a pair of coordinate axes. Make sure you use a large area of your page to draw the graph – at least half a page. Once the points are plotted, drawing the graph of the parabola is a matter of joining the dots with a nice smooth bowl-shaped curve. Remember that the curve turns around at the vertex, and the vertical line through the vertex is the axis of symmetry of the curve.
You can check your graph using software like GraphPower.