Which shape do we get from a Quadratic Equation?

When we graph a quadratic equation of the form \(y=ax^2 +bx +c\), we get a parabola (a U-shaped curve going up or down).

To graph any given quadratic equation, follow the 6 steps below.


Step 1: Identify the coefficients a, b, c.

If the equation is not written in standard form, we need to first turn it into standard form 
\(y=ax^2 +bx +c\)

where \(a, b \) and \(c\) are coefficients (constants) and \(x, y\) are variables. 

By doing this, we identify the values of \(a, b,\) and \(c\), which we will use in the following steps. 


Step 2: Determine the Vertex + direction of the Parabola

The vertex is the lowest or highest point of the parabola. If \(a\) is positive, then the parabola goes up and the vertex is the lowest point (smallest value of \(y\)).


If \(a\) is negative, the parabola goes down and the vertex is the highest point (biggest value of \(y\)).



The x-coordinate of the vertex of the quadratic equation is:
 \(x=-\frac{b}{2a}\)


Once we have this value of x, we subsititute it in the given equation, and find \(y\).

Mark this point (the vertex) on the coordinate plane. 

Notice we didn't talk about when \(a=0\)? Why do you think that is?

Step 3: Determine the axis of symmetry

The axis of symmetry is a vertical line that passes through the vertex and its equation is: \(x=-\frac{b}{2a}\)

Step 4: Determine the x and y-intercepts

It is important to understand where the parabola crosses (touches) the x-axis and y-axis.

Note: There are cases where the parabola does not cross one or both axes and in those cases you will proceed to the next step. To find the point where the parabola crosses the x-axis, set \(y = 0\), and solve for \(x\). To find the y-intercept, we set the \(x=0\), and solve for \(y\).

Step 5: Add additional points.

Depending on the equation and level of accuracy required for your graph, you might need to add more points to both sides of the axis of symmetry (step 3). To do this, choose an \(x\) that is smaller, and an \(x\) that is bigger than the \(x\) in step 3, and solve for \(y\).

Step 6: Connect all the dots from steps 2-5.

Your graph is ready. Make sure you mark all the points from previous steps and remember that we are graphing a curve. Don't bring out your rulers!