Algebra unit 10.2

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Unit 10.2

Transcript of Algebra unit 10.2

  • UNIT 10.2 QUADRATIC FUNCTIONS

  • Warm Up

    Find the axis of symmetry.1. y = 4x2 72. y = x2 3x + 1 3. y = 2x2 + 4x + 34. y = 2x2 + 3x 1

    Find the vertex.5. y = x2 + 4x + 56. y = 3x2 + 2 7. y = 2x2 + 2x 8 x = 0 x = 1 (2, 1) (0, 2)

  • Graph a quadratic function in the form y = ax2 + bx + c.Objective

  • Recall that a y-intercept is the y-coordinate of the point where a graph intersects the y-axis. The x-coordinate of this point is always 0. For a quadratic function written in the form y = ax2 + bx + c, when x = 0, y = c. So the y-intercept of a quadratic function is c.

  • Example 1: Graphing a Quadratic Function Graph y = 3x2 6x + 1.Step 1 Find the axis of symmetry.= 1 The axis of symmetry is x = 1. Simplify.Step 2 Find the vertex.y = 3x2 6x + 1 = 3(1)2 6(1) + 1= 3 6 + 1= 2The vertex is (1, 2). The x-coordinate of the vertex is 1. Substitute 1 for x.Simplify.The y-coordinate is 2.

  • Example 1 Continued Step 3 Find the y-intercept.y = 3x2 6x + 1y = 3x2 6x + 1The y-intercept is 1; the graph passes through (0, 1).Identify c.

  • Step 4 Find two more points on the same side of the axis of symmetry as the point containing the y-intercept.Since the axis of symmetry is x = 1, choose x-values less than 1.Let x = 1.y = 3(1)2 6(1) + 1 = 3 + 6 + 1 = 10Let x = 2. y = 3(2)2 6(2) + 1 = 12 + 12 + 1 = 25Substitutex-coordinates.Simplify.Two other points are (1, 10) and (2, 25). Example 1 Continued

  • Graph y = 3x2 6x + 1.Step 5 Graph the axis of symmetry, the vertex, the point containing the y-intercept, and two other points.Step 6 Reflect the points across the axis of symmetry. Connect the points with a smooth curve.Example 1 Continued

  • Check It Out! Example 1a Graph the quadratic function. y = 2x2 + 6x + 2Step 1 Find the axis of symmetry.Simplify.

  • Step 2 Find the vertex.y = 2x2 + 6x + 2Simplify.Check It Out! Example 1a Continued

  • Step 3 Find the y-intercept.y = 2x2 + 6x + 2y = 2x2 + 6x + 2The y-intercept is 2; the graph passes through (0, 2).Identify c.Check It Out! Example 1a Continued

  • Step 4 Find two more points on the same side of the axis of symmetry as the point containing the y-intercept.Let x = 1y = 2(1)2 + 6(1) + 1 = 2 6 + 2 = 2Let x = 1 y = 2(1)2 + 6(1) + 2 = 2 + 6 + 2 = 10Substitutex-coordinates.Simplify.Two other points are (1, 2) and (1, 10). Check It Out! Example 1a Continued

  • Step 5 Graph the axis of symmetry, the vertex, the point containing the y-intercept, and two other points.Step 6 Reflect the points across the axis of symmetry. Connect the points with a smooth curve.y = 2x2 + 6x + 2Check It Out! Example 1a Continued

  • Check It Out! Example 1b Graph the quadratic function. y + 6x = x2 + 9Step 1 Find the axis of symmetry.Simplify.The axis of symmetry is x = 3. = 3y = x2 6x + 9Rewrite in standard form.

  • Step 2 Find the vertex.Simplify.Check It Out! Example 1b Continued = 9 18 + 9= 0The vertex is (3, 0). The x-coordinate of the vertex is 3. Substitute 3 for x.The y-coordinate is 0. .y = x2 6x + 9y = 32 6(3) + 9

  • Step 3 Find the y-intercept.y = x2 6x + 9y = x2 6x + 9The y-intercept is 9; the graph passes through (0, 9).Identify c.Check It Out! Example 1b Continued

  • Step 4 Find two more points on the same side of the axis of symmetry as the point containing the y- intercept.Since the axis of symmetry is x = 3, choose x-values less than 3.Let x = 2y = 1(2)2 6(2) + 9 = 4 12 + 9 = 1Let x = 1 y = 1(1)2 6(1) + 9 = 1 6 + 9 = 4Substitutex-coordinates.Simplify.Two other points are (2, 1) and (1, 4). Check It Out! Example 1b Continued

  • Step 5 Graph the axis of symmetry, the vertex, the point containing the y-intercept, and two other points.Step 6 Reflect the points across the axis of symmetry. Connect the points with a smooth curve.y = x2 6x + 9Check It Out! Example 1b Continued

  • Example 2: ApplicationThe height in feet of a basketball that is thrown can be modeled by f(x) = 16x2 + 32x, where x is the time in seconds after it is thrown. Find the basketballs maximum height and the time it takes the basketball to reach this height. Then find how long the basketball is in the air.

  • Example 2 ContinuedThe answer includes three parts: the maximum height, the time to reach the maximum height, and the time to reach the ground.The function f(x) = 16x2 + 32x models the height of the basketball after x seconds. List the important information:

  • Find the vertex of the graph because the maximum height of the basketball and the time it takes to reach it are the coordinates of the vertex. The basketball will hit the ground when its height is 0, so find the zeros of the function. You can do this by graphing. Example 2 Continued

  • Step 1 Find the axis of symmetry.Simplify.The axis of symmetry is x = 1.Example 2 Continued

  • Step 2 Find the vertex.f(x) = 16x2 + 32x= 16(1)2 + 32(1)= 16(1) + 32= 16 + 32= 16The vertex is (1, 16).The x-coordinate of the vertex is 1. Substitute 1 for x.Simplify.The y-coordinate is 16.Example 2 Continued

  • Step 3 Find the y-intercept.Identify c.f(x) = 16x2 + 32x + 0The y-intercept is 0; the graph passes through (0, 0).Example 2 Continued

  • Step 4 Graph the axis of symmetry, the vertex, and the point containing the y-intercept. Then reflect the point across the axis of symmetry. Connect the points with a smooth curve.Example 2 Continued

  • The vertex is (1, 16). So at 1 second, the basketball has reached its maximum height of 16 feet. The graph shows the zeros of the function are 0 and 2. At 0 seconds the basketball has not yet been thrown, and at 2 seconds it reaches the ground. The basketball is in the air for 2 seconds.Example 2 Continued

  • Look BackCheck by substitution (1, 16) and (2, 0) into the function.Example 2 Continued

  • Check It Out! Example 2 As Molly dives into her pool, her height in feet above the water can be modeled by the function f(x) = 16x2 + 24x, where x is the time in seconds after she begins diving. Find the maximum height of her dive and the time it takes Molly to reach this height. Then find how long it takes her to reach the pool.

  • The answer includes three parts: the maximum height, the time to reach the maximum height, and the time to reach the pool.Check It Out! Example 2 ContinuedList the important information:The function f(x) = 16x2 + 24x models the height of the dive after x seconds.

  • Find the vertex of the graph because the maximum height of the dive and the time it takes to reach it are the coordinates of the vertex. The diver will hit the water when its height is 0, so find the zeros of the function. You can do this by graphing. Check It Out! Example 2 Continued

  • Step 1 Find the axis of symmetry.Simplify.The axis of symmetry is x = 0.75.Check It Out! Example 2 Continued

  • Step 2 Find the vertex.f(x) = 16x2 + 24x= 16(0.75)2 + 24(0.75)= 16(0.5625) + 18= 9 + 18= 9The vertex is (0.75, 9).Simplify.The y-coordinate is 9.Check It Out! Example 2 Continued

  • Step 3 Find the y-intercept.Identify c.f(x) = 16x2 + 24x + 0The y-intercept is 0; the graph passes through (0, 0).Check It Out! Example 2 Continued

  • Step 4 Find another point on the same side of the axis of symmetry as the point containing the y-intercept.Since the axis of symmetry is x = 0.75, choose an x-value that is less than 0.75.Let x = 0.5f(x) = 16(0.5)2 + 24(0.5) = 4 + 12= 8 Another point is (0.5, 8).Substitute 0.5 for x.Simplify.Check It Out! Example 2 Continued

  • Step 5 Graph the axis of symmetry, the vertex, the point containing the y-intercept, and the other point. Then reflect the points across the axis of symmetry. Connect the points with a smooth curve.Check It Out! Example 2 Continued

  • The vertex is (0.75, 9). So at 0.75 seconds, Molly's dive has reached its maximum height of 9 feet. The graph shows the zeros of the function are 0 and 1.5. At 0 seconds the dive has not begun, and at 1.5 seconds she reaches the pool. Molly reaches the pool in 1.5 seconds.Check It Out! Example 2 Continued

  • Look BackCheck by substitution (0.75, 9) and (1.5, 0) into the function.Check It Out! Example 2 Continued

  • Lesson Quiz1. Graph y = 2x2 8x + 4.

    2. The height in feet of a fireworks shell can be modeled by h(t) = 16t2 + 224t, where t is the time in seconds after it is fired. Find the maximum height of the shell, the time it takes to reach its maximum height, and length of time the shell is in the air.

    784 ft; 7 s; 14 s

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