Test Questions for Graphing Systems of Non-Linear Inequalities (Algebra 2)

📚 Quick Study Guide

• 📈 Inequality Basics: Inequalities use symbols like $<$, $>$, $\leq$, and $\geq$ to represent regions on a graph rather than just lines.
• 🎨 Graphing Non-Linear Inequalities: Replace the inequality sign with an equals sign and graph the resulting curve (e.g., parabola, circle, hyperbola). Use a dashed line for strict inequalities ($<$ or $>$) and a solid line for inclusive inequalities ($\leq$ or $\geq$).
• 🧪 Test Points: Choose a test point NOT on the curve. Substitute the point's coordinates into the original inequality. If the inequality holds true, shade the region containing the test point; otherwise, shade the opposite region.
• 🎯 System of Inequalities: The solution to a system of inequalities is the region where the shaded areas of all inequalities overlap.
• 💡 Common Non-Linear Equations:
  • Circle: $(x-h)^2 + (y-k)^2 = r^2$
  • Parabola: $y = ax^2 + bx + c$ or $x = ay^2 + by + c$
  • Ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
  • Hyperbola: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ or $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$

Practice Quiz

1. Which inequality represents the area inside a circle with radius 3 centered at the origin?
   1. $x^2 + y^2 > 9$
   2. $x^2 + y^2 < 9$
   3. $x^2 + y^2 \geq 9$
   4. $x^2 + y^2 \leq 9$
2. Which of the following is the correct way to graph $y > x^2 + 2$?
   1. Draw a solid parabola opening upwards and shade the region below the curve.
   2. Draw a dashed parabola opening upwards and shade the region above the curve.
   3. Draw a solid parabola opening upwards and shade the region above the curve.
   4. Draw a dashed parabola opening upwards and shade the region below the curve.
3. Which point satisfies the system of inequalities: $y \geq x^2$ and $y < -x + 2$?
   1. (0, 3)
   2. (1, 1)
   3. (2, 4)
   4. (-1, 0)
4. Which inequality represents the area outside the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$?
   1. $\frac{x^2}{4} + \frac{y^2}{9} < 1$
   2. $\frac{x^2}{4} + \frac{y^2}{9} \leq 1$
   3. $\frac{x^2}{4} + \frac{y^2}{9} > 1$
   4. $\frac{x^2}{4} + \frac{y^2}{9} \geq 1$
5. What type of line should be used to graph the inequality $y \leq x^2 - 1$?
   1. Dashed line
   2. Solid line
   3. Dotted line
   4. No line
6. Which region should be shaded for the inequality $x^2 + y^2 > 16$?
   1. Inside the circle with radius 4
   2. Outside the circle with radius 4
   3. On the circle with radius 4
   4. Only the x-axis
7. Which of the following systems of inequalities has no solution?
   1. $y > x^2$ and $y < x^2 + 1$
   2. $y \geq x^2$ and $y \leq -x^2$
   3. $y > x^2$ and $y < -x^2$
   4. $y < x^2$ and $y > -x^2 - 1$