Which shape do we get from a Quadratic Equation?

When we graph a quadratic equation of the form \(y=a(x-h)^2 +k \), we get a parabola (a U-shaped curve).

To graph any given quadratic equation, follow the 5 steps below.


Step 1: Identify the vertex coordinates (h,k)

In the equation \(y=a(x-h)^2 + k\), the numbers \(h\) and \(k\) are the coordinates of the vertex. In this first step, we can add the vertex (\(h,k\)) to the graph.

Next, we need to know if the parabola is going up or down.

Direction?

  • Is the parabola going up?

    If \(a\) is a positive number, then the parabola goes up and the vertex is the lowest point in the graph.

  • Or down?

    If \(a\) is negative, then the parabola will go down and the vertex will be the highest point in the graph.

Step 2: Determine the axis of symmetry

The axis of symmetry is a vertical line, paralel to the y-axis, that passes through the vertex. It is given by the equation: \(x=h\). Remember that \(h\) is the \(x\) coordinate of the vertex (step 1). This line will help us "mirror" the points that we find in steps 3 and 4.

Step 3: 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 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 \(y\). We add all these points to the graph. We can start to see where our parabola will be.

Step 4: 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 2). To do this, choose an \(x\) that is smaller, and an \(x\) that is bigger than the \(x\) in step 3, and solve for \(y\). By using the axis of symmetry, we can "mirror" these new points to the opposite side.

Step 5: Connect all the dots from steps 1- 4.

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!