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6-7 Graphing and Solving Quadratic Inequalities
Objectives: 1) Graph Quadratic Inequalities in Two Variables. 2) Solve Quadratic Inequalities in One Variable.
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Recall how to Graph Linear Inequalities:
y < -3x + 5
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Graphing Quadratic Inequalities
You can graph quadratic inequalities in two variables using the same techniques you used to graph linear inequalities in two variables. Steps: Determine whether the boundary should be solid or dashed. Graph the quadratic function. Test a point inside the parabola. Check to see if this ordered pair is a solution to the inequality. If the resulting Inequality is TRUE, shade INSIDE the parabola. If the resulting inequality is FALSE, shade OUTSIDE your parabola
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Example 1a: Graph a Quadratic Inequality
Graph y < x2 -6x – 7 Methods to graph: Table of values. Find x-intercepts (zeros) and vertex.
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Example 1b: Graph a Quadratic Inequality
Graph y > x2 - 4
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Example 1c: Graph a Quadratic Inequality
Graph y > - x2 + 10x - 25
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Solving Quadratic Inequalities
To solve a quadratic inequality in one variable, you can use the graph of the related quadratic function. To solve ax2 + bx + c < 0, graph y = ax2 + bx + c. Identify the x values for which the graph lies BELOW the x-axis. To solve ax2 + bx + c > 0, graph y = ax2 + bx + c. Identify the x values for which the graph lies ABOVE the x-axis. For < or >, include the x-intercepts in the solution.
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Example 2a: Solve by Graphing.
Solve x2 + 2x – 3 > 0
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Example 2b: Solve by Graphing.
Solve –x2 – 10x – 21 < 0
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Example 2c: Solve by Graphing.
Solve x2 – 9 > 0
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Solve Quadratic Inequalities Algebraically
Solve the related equation to identify the zeros. Plot the zeros on a number line (be careful with open vs. closed circles). Test a value in each interval to see if it satisfies the original inequality.
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Example 3a: Solve Algebraically
Solve x2 + x > 6
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Example 3b: Solve Algebraically
Solve x2 - 4x < 5
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You Try It… Solve the quadratic inequality algebraically. 9x < 12x2
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Application Vanessa has 180 feet of fencing that she intends to use to build a rectangular play area for her dog. She wants the play area to enclose at least 1800 square feet. What are the possible widths of the play area?
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Homework Text p #s 1-8 all Text p. 333 #s 9-13 all
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