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Lesson Menu Five-Minute Check (over Lesson 4–7) CCSS Then/Now New Vocabulary Example 1:Graph a Quadratic Inequality Example 2:Solve ax 2 + bx + c < 0 by Graphing Example 3:Solve ax 2 + bx + c ≥ 0 by Graphing Example 4:Real-World Example: Solve a Quadratic Inequality Example 5:Solve a Quadratic Inequality Algebraically

Over Lesson 4–7 5-Minute Check 1 Rewrite the equation in vertex form. Identify the vertex. Find the axis of symmetry. Which way does the parabola open? Graph the function. The graph of y = x 2 is reflected in the x-axis and shifted left three units. What is the resulting equation in vertex form? (1-5) Consider y = 5x x + 44

Over Lesson 4–7 5-Minute Check 1 A.y = x 2 + 6x + 11 B.y = 5(x + 3) 2 – 1 C.y = 5(x – 3) 2 D.y = (x + 3) Write y = 5x x + 44 in vertex form.

Over Lesson 4–7 5-Minute Check 2 A.(3, 1) B.(–1, –3) C.(–2, –1) D.(–3, –1) Identify the vertex of y = 5x x + 44.

Over Lesson 4–7 5-Minute Check 3 A.x = 3 B.x = 0 C.x = –2 D.x = –3 Find the axis of symmetry of y = 5x x + 44.

Over Lesson 4–7 5-Minute Check 4 A.upward B.downward C.right D.left What is the direction of the opening of the graph of y = 5x x + 44?

Over Lesson 4–7 5-Minute Check 5 Graph the quadratic function y = 5x x A. C. B. D.

Over Lesson 4–7 5-Minute Check 6 A.y = (x – 3) 2 B.y = –(x – 3) 2 C.y = –(x + 3) 2 D.y = –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?

CCSS Content Standards A.CED.1 Create equations and inequalities in one variable and use them to solve problems. A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Mathematical Practices 1 Make sense of problems and persevere in solving them.

Then/Now You solved linear inequalities. Graph quadratic inequalities in two variables. Solve quadratic inequalities in one variable.

Vocabulary quadratic inequality

Vocabulary Graphing Procedure Three Steps: 1.Graph equality 2.Use a test point 3.Ask: “Is the line included?”

Example 1 Graph a Quadratic Inequality Graph y > x 2 – 3x + 2. Step 1 Graph y = x 2 – 3x + 2. xx 2 - 3x + 2y 0(0) 2 - 3(0) (.5) 2 - 3(.5) (1) 2 - 3(1) (1.5) 2 - 3(1.5) (2) 2 - 3(2) (2.5) 2 - 3(2.5) (3) 2 - 3(3) + 22

Example 1 Graph a Quadratic Inequality Graph y > x 2 – 3x + 2. Step 1 Graph y = x 2 – 3x + 2. Step 2 Use a test point. Consider (0,0) Notice

Example 1 Graph a Quadratic Inequality Graph y > x 2 – 3x + 2. Step 1 Graph y = x 2 – 3x + 2. Step 2 Use a test point. Consider (0,0) Step 3 Ask “Line included?” “>” so line is NOT included

A.B. C.D. Example 1 Which is the graph of y < –x 2 + 4x + 2?

Example 2 Solve ax 2 + bx + c < 0 by Graphing Solve x 2 – 4x + 3 < 0 by graphing. Graph y = x 2 - 4x + 3 Note: y = (x – 2) Find the x values that correspond to negative y values. Answer: The solution set is {x | 1 < x < 3}.

Example 2 A.{x | –3 < x < –2} B.{x | x –2} C.{x | 2 < x < 3} D.{x | x 3} What is the solution to the inequality x 2 + 5x + 6 < 0?

Example 3 Solve ax 2 + bx + c ≥ 0 by Graphing Solve 0 ≤ –2x 2 – 6x + 1 by graphing.

Example 3 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} Solve 2x 2 + 3x – 7 ≥ 0 by graphing.

Example 4 Solve a Quadratic Inequality SPORTS The height of a ball above the ground after it is thrown upwards at 40 feet per second can be modeled by the function h(t) = 40t – 16t 2, where the height h(t) is given in feet and the time t is in seconds. At what time is the ball within 15 feet of the ground? Find values for which h(t) ≤ 15. h(t)≤15 40t – 16t 2 ≤15 –16t t – 15≤0 Zeros are t ≈ 0.46 and t ≈ 2.04 sec. Answer: The ball is within 15 feet of the ground for the first 0.46 second of its flight, from 0 to 0.46 second, and again after 2.04 seconds until it hits the ground at 2.5 seconds.

Example 4 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. SPORTS 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(t) = 28x – 16x 2, where the height h(t) is given in feet and the time t is in seconds. At what time is the ball within 10 feet of the ground?

Example 5 Solve a Quadratic Inequality Algebraically Solve x 2 + x ≤ 2 algebraically. Solve the related quadratic equation x 2 + x = 2. Graph the solutions x 2 + x=2 x 2 + x – 2=0 (x + 2)(x – 1)=0 x + 2 = 0 or x – 1 = 0 x=–2 or x= x+2 1 x-1 x 2 +x Answer: The solution set is {x | –2 ≤ x ≤ 1}. This is shown on the number line below.

Example 5 A.{x | –3 ≤ x ≤ –2} B.{x | x ≥ –2 or x ≤ –3} C.{x | 1 ≤ x ≤ 6} D.{x | –6 ≤ x ≤ –1} Solve x 2 + 5x ≤ –6 algebraically. What is the solution?

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