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Graphing Quadratic Inequalities

Graphing Quadratic Inequalities

Author: Sophia Tutorial

This lesson covers graphing quadratic inequalities.

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What's Covered

  • Graphing a Parabola
  • Graphing an Inequality
  • Graphing a Quadratic Inequality

Graphing Quadratic Inequalities

Graphing a Parabola

More than likely, you'll be given a quadratic equation or inequality in standard form: ax2 + bx + c.  To graph this, it is helpful to find the vertex to the parabola, and then plot some points on one side of the vertex.  Since the vertex lies on the axis of symmetry, we can simply reflect the already plotted points across this line, to create a more complete sketch of the parabola. 

To find the vertex and/or axis of symmetry when working in standard form, we use the formula: x equals negative fraction numerator b over denominator 2 a end fraction, where a and b come from the coefficients of the x-squared and x-terms in standard form. 

The previous formula is the equation to the vertical line which is the axis of symmetry (for vertical parabolas).  This also represents the x–coordinate of the vertex.  To find the y–coordinate, simply use the calculated value of x in the original equation, and solve for y.  


Graphing Inequalities

Graphing inequalities is similar to graphing equations, but there are some important differences.  First, there is a difference between using a solid line and a dashed line to draw the curve.  If the inequality symbol is strict ( < or >) we always use a dashed line to show that we are not including exact values on the boundary line.  If the inequality symbol is non-strict ( ≤ or ≥ ) we always use a solid line to show that we are including exact values on the boundary line. 

Not only do we need to be careful about what type of line to use when drawing the graph, we also need to highlight a solution region.  The highlighted solution region represents the set of all (x, y) coordinates that satisfy the inequality.  To decide which inequality region to highlight, we use a test point.  This test point can be anywhere on the coordinate plane, so long as it isn't on the boundary line (because that would not help us at all).  We take the x– and y– values from this test point and plug it into our inequality.  If the statement is true, the half-plane that the test point lies within is the solution region, and gets highlighted.  If not, the other half-plane gets highlighted. 

Graphing a Quadratic Inequality

Let's apply what we know about graphing parabolas and graphing inequalities to graph a quadratic inequality. Let's graph the following inequality:

y greater or equal than x squared minus 2 x plus 2

The first thing we need to do is graph the boundary line, y = x2 – 2x + 2.  We will use a solid line to graph the parabola, because our inequality symbol is non-strict. To graph the boundary line, we need to find the vertex, plot a couple of points, and then reflect them across the axis of symmetry: 

We use x = 1 to find the y–coordinate of the vertex: 

Next, we plot a few points on one side of the vertex, and reflect them across the axis of symmetry to sketch the boundary line: 

The final step is to choose any test point (not on the boundary line) to plug x– and y–values into the inequality.  Remember, a true statement means the point lies within the solution region (and that half-plane should be shaded).  Otherwise, the other half-plane gets shaded. 


Use the test point (0, 0) if you can.  This is because it is easy to multiply by zero and add zero to quantities.  It is always recommended to make calculations simpler!

Since the point (0, 0) yields a false statement, we highlight the other region of the coordinate plane as our solution region (we do not highlight the region that includes the origin (0,0) in this case):