1. ALGEBRA 1 Lesson 9-2 Warm-Up

2. ALGEBRA 1 “Quadratic Functions” (9-2) How do you find a parabola’s axis of symmetry (fold or line that divides the parabola into two matching halves). Recall that the standard form of a quadratic function is ax 2 + bx + c and that the graph of a quadratic function is a parabola (U-shaped curve). The position of a parabola’s axis of symmetry (fold or line that divides the parabola into two matching halves) is based on a ratio between the b and a values (the c value is the y-intercept). Since the parabola of a quadratic function open upward or downward, the axis of symmetry is a vertical line, which means it is somewhere along the x axis. Rule: Axis of Symmetry: The axis of symmetry, x, of a function in the form of ax 2 + bx + c is: x (axis of symmetry) = – = – Examples: y = 2x 2 + 2x – = - = - In the graph, notice that there is no y-intercept, since there is no c value. S baba 1212 b 2a b2ab2a 2 2 (2) 1 2

3. ALGEBRA 1 “Quadratic Functions” (9-2) How do you graph a quadratic function? To graph a quadratic function, you will need to find the vertex, the axis of symmetry, and at least two points on each side of the axis of symmetry. Step 1: Find the axis of symmetry and the vertex. The axis of symmetry is x = 1. The vertex is (1, 8) Step 2: Find two other points on one side of the line of symmetry (if possible, one should be the y-intercept) When x = -1, y = -4, so another point is (-1, -4). S

4. ALGEBRA 1 “Quadratic Functions” (9-2) Step 3: Reflect the points you found (graph the point that is the same distance on the opposite side of the line of symmetry)

5. ALGEBRA 1 Graph the function y = 2x 2 + 4x – 3. Step 1: Find the axis of symmetry and the coordinates of the vertex. Find the equation of the axis of symmetry.x = b2ab2a – = –4 2(2) = – 1 The axis of symmetry is x = –1. The vertex is (–1, –5). y = 2x 2 + 4x – 3 To find the y-coordinate of the vertex, substitute –1 for x. y = 2(–1) 2 + 4(–1) – 3 = –5 Quadratic Functions LESSON 9-2 Additional Examples

6. ALGEBRA 1 (continued) Step 2: Find two other points on the graph. For x = 0, y = –3, so one point is (0, –3). Use the y-intercept. Choose a value for x on the same side of the vertex. Let x = 1 Find the y-coordinate for x = 1. y = 2(1) 2 + 4(1) – 3 = 3 For x = 1, y = 3, so another point is (1, 3). Quadratic Functions LESSON 9-2 Additional Examples

7. ALGEBRA 1 (continued) Step 3: Reflect (0, –3) and (1, 3) across the axis of symmetry to get two more points. The domain is the set of all real numbers. The range is {y : y ≥ –5}. Quadratic Functions LESSON 9-2 Additional Examples Then draw the parabola.

8. ALGEBRA 1 Aerial fireworks carry “stars,” which are made of a sparkler-like material, upward, ignite them, and project them into the air in fireworks displays. Suppose a particular star is projected from an aerial firework at a starting height of 610 ft with an initial upward velocity of 88 ft/s. How long will it take for the star to reach its maximum height? How far above the ground will it be? The equation h = –16t 2 + 88t + 610 gives the height of the star h in feet at time t in seconds. Since the coefficient of t 2 is negative, the curve opens downward, and the vertex is the maximum point. Quadratic Functions LESSON 9-2 Additional Examples

9. ALGEBRA 1 (continued) Step 2: Find the h-coordinate of the vertex. The maximum height of the star will be about 731 ft. Step 1: Find the x-coordinate of the vertex. b2ab2a – = –88 2(–16) = 2.75 After 2.75 seconds, the star will be at its greatest height. h = –16(2.75) 2 + 88(2.75) + 610Substitute 2.75 for t. h = 731Simplify using a calculator. Quadratic Functions LESSON 9-2 Additional Examples

10. ALGEBRA 1 “Quadratic Functions” (9-2) How do you graph a quadratic inequality. Graph a quadratic inequality is similar to graphing a linear inequality. The parabola becomes the boundary line separating solutions from non-solutions. Just like a linear inequality, the curve (boundary line) is dashed if the inequality involves  or  and solid if the inequality involves ≥ or ≤. To figure out which side of the curve is shaded, test a point on each side to see if it is a solution of the inequality (in other words, makes the inequality a true statement). Example: Graph y ≤ x 2 - 3x – 4 Step 1: Graph the inequality. Use a solid line, because the inequality include the boundary line with ≤ (less than or equal) Step 2: Check points on each side of the curve. Shade in the side in which the tested point makes the inequality true.. Test a point inside the curve, like (0, 0) y ≤ x 2 - 3x – 4 ; 0 ≤ 0 2 – 3(0) - 4 0 ≤ - 4 (not a true statement) Test a point outside the curve, like (5, 0) y ≤ x 2 - 3x – 4 ; 0 ≤ 5 2 – 3(5) - 4 0 ≤ 25 – 15 – 4; 0 ≤ 6 0 ≤ 6 (true statement)  S

11. ALGEBRA 1 Graph the boundary curve, y = –x 2 + 6x – 5. Use a dashed line because the solution of the inequality y > –x 2 + 6x – 5 does not include the boundary. Graph the quadratic inequality y > –x 2 + 6x – 5. Shade above the curve. Quadratic Functions LESSON 9-2 Additional Examples

12. ALGEBRA 1 Graph each relation. Label the axis of symmetry and the vertex. Find the domain and range. 2.ƒ(x) = –x 2 + 4x – 21.y = x 2 – 8x + 15 domain: set of all real numbers; range: {y : y ≥ –1} domain: set of all real numbers; range: {y : y ≤2} Quadratic Functions LESSON 9-2 Lesson Quiz

13. ALGEBRA 1 1414 < 3.y – x 2 – 2x – 6 x = –4 domain: set of all real numbers; range: {y : y ≤ – 2} Quadratic Functions LESSON 9-2 Lesson Quiz