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Graphing Quadratic Inequalities

Graphing Quadratic Inequalities 1.9 - Solution

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To graph the quadratic inequality, we will follow three steps.

  1. Graph the related quadratic function.
  2. Test a point not on the parabola.
  3. Shade accordingly. If the point satisfies the inequality, we shade the region that contains the point. If not, we shade the opposite region.

Step 11

Let's draw the graph of the related function, which is y=x28x+2.y=x^2-8x+2.

Step 22

Next, let's determine which region to shade by testing a point. For simplicity, we will use (0,0)(0,0) as our test point. Let's see if it satisfies the given inequality.
yx28x+2y\leq x^2-8x+2
0?(0)28(0)+2{\color{#009600}{0}}\stackrel{?}{\leq} ({\color{#0000FF}{0}})^2-8({\color{#0000FF}{0}})+2
0?08(0)+20\stackrel{?}{\leq} 0-8(0)+2
0?00+20\stackrel{?}{\leq} 0-0+2
02 0\leq 2 \ {\color{#009600}{\huge\checkmark}}

Step 33

Since (0,0)(0,0) produced a true statement, we will shade the region that contains the point. Also, note that the inequality is not strict. Therefore, the parabola will be solid.