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Name:__________ warm-up 4-8 Write y = 2x x + 44 in vertex formIdentify the vertex of y = 5x x Find the axis of symmetry of y = 5x x Find the axis of symmetry of y = 5x x + 44.

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What is the direction of the opening of the graph of y = 5x x + 44? Graph the quadratic function y = 5x x The graph of y = x 2 is reflected in the x-axis and shifted left three units. Which of the following equations represents the resulting parabola? A.y = (x – 3) 2 B.y = –(x – 3) 2 C.y = –(x + 3) 2 D.y = –x A. C. B. D.

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Name:__________ warm-up Review Write y = y = 2x x + 25 in vertex formIdentify the vertex of y = 2x x Find the axis of symmetry of y = 2x x Find the axis of symmetry of y = 2x x + 25.

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What is the direction of the opening of the graph of y = 2x x + 25? Graph the quadratic function y = 2x x + 25 Solve x 2 - 3x -10 ˂ 0 algebraically and then graph.

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Details of the Day EQ: How do quadratic relations model real-world problems and their solutions? Depending on the situation, why is one method for solving a quadratic equation more beneficial than another? How do transformations help you to graph all functions? Why do we need another number set? I will be able to… Activities: Warm-up Review homework – Notes: 4-8 Quadratic Inequalities Class work/ HW Vocabulary: quadratic inequality. Graph quadratic inequalities in two variables. Solve quadratic inequalities in one variable. https://www.youtube.com/watch?v=BXWzNp-4ilA https://www.youtube.com/watch?v=t54ccHYVhoo https://www.youtube.com/watch?v=8FCV2-EjgXg https://www.mathsisfun.com/algebra/inequality-quadratic- solving.html

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Details of the Day EQ: How do quadratic relations model real-world problems and their solutions? Depending on the situation, why is one method for solving a quadratic equation more beneficial than another? How do transformations help you to graph all functions? Why do we need another number set? I will be able to… Activities: Warm-up- vocabulary self quiz Review homework – Round robin student review Student presentations. Graph quadratic inequalities in two variables. Solve quadratic inequalities in one variable.

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4-8 Quadratic Inequalities

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A Quick Review Write y = 5x x + 44 in vertex formIdentify the vertex of y = 5x x Find the axis of symmetry of y = 5x x Find the axis of symmetry of y = 5x x + 44.

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A Quick Review What is the direction of the opening of the graph of y = 5x x + 44? Graph the quadratic function y = 5x x The graph of y = x 2 is reflected in the x-axis and shifted left three units. Which of the following equations represents the resulting parabola? A.y = (x – 3) 2 B.y = –(x – 3) 2 C.y = –(x + 3) 2 D.y = –x A. C. B. D.

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Notes and examples Graph the related quadratic equation, y = x 2 – 3x + 2. Since the inequality symbol is >, the parabola should be dashed. Test a point inside the parabola, such as (1, 2). Shade the region

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Notes and examples graph of y < –x 2 + 4x + 2

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Notes and examples Solve x 2 – 4x + 3 < 0 by graphing. The solution consists of the x-values for which the graph of the related quadratic function lies below the x-axis. Begin by finding the roots of the related equation. Related Factor Zero Product Property Solve each equation

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Notes and examples Sketch the graph of the parabola that has x-intercepts at 3 and 1. The graph should open up since a > 0. The graph lies below the x-axis to the right of x = 1 and to the left of x = 3.

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Notes and examples What is the solution to the inequality x 2 + 5x + 6 < 0?

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Notes and examples Solve 0 ≤ –2x 2 – 6x + 1 by graphing This inequality can be rewritten as –2x 2 – 6x + 1 ≥ 0. The solution consists of the x-values for which the graph of the related quadratic equation lies on and above the x-axis. Begin by finding roots of the related equation Sketch the graph of the parabola Write Equation Write quadratic equation Substitute Simplify

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Notes and examples Test one value of x less than –3.16, one between –3.16 and 0.16, and one greater than 0.16 in the original inequality.

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Notes and examples Solve 2x 2 + 3x – 7 ≥ 0 by graphing. A.{x | –2.77 ≤ x ≤ 1.27} B.{x | –1.27 ≤ x ≤ 2.77} C.{x | x ≤ –2.77 or x ≥ 1.27} D.{x | x ≤ –1.27 or x ≥ 2.77}

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Notes and examples The height of a ball above the ground after it is thrown upwards at 28 feet per second can be modeled by the function h(x) = 28x – 16x 2, where the height h(x) is given in feet and the time x is in seconds. At what time in its flight is the ball within 10 feet of the ground? h(x)≤15 Original inequality Graph the related function using a graphing calculator.

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Notes and examples The height of a ball above the ground after it is thrown upwards at 28 feet per second can be modeled by the function h(x) = 28x – 16x 2, where the height h(x) is given in feet and the time x is in seconds. At what time in its flight is the ball within 10 feet of the ground? A.for the first 0.5 second and again after 1.25 seconds B.for the first 0.5 second only C.between 0.5 second and 1.25 seconds D.It is never within 10 feet of the ground.

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Notes and examples Solve x 2 + x ≤ 2 algebraically First, solve the related quadratic equation x 2 + x = 2. Test a value in each interval to see if it satisfies the original inequality Plot points on a number line. Use closed circles since these solutions are included. Notice that the number line is separated into 3 interval

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Notes and examples Solve x 2 + 5x ≤ –6 algebraically. What is the solution?

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