2. Course: Adv. Alg. & Trig. Aim: Graphing Parabola Do Now: Aim: How do we graph a parabola?

3. Course: Adv. Alg. & Trig. Aim: Graphing Parabola Graphing the parabola y = x 2 Table of Values Graph y = x 2 for the values -3 < x < 3 (-3) 2 9 -3,9 (-2) 2 4 -2,4 (-1) 2 1 -1,1 (0) 2 0 0,0 (1) 2 1 1,1 (2) 2 4 2,4 (3) 2 9 3,9 -1,1 (0,0) (-2,4) (-3,9) (1,1) (2,4) (3,9) y = x 2 Axis of symmetry- Turning point- x-intercept & y-intercept - x = 0 (0,0) Minimum (0,0) x y

4. Course: Adv. Alg. & Trig. Aim: Graphing Parabola Graphing the parabola y = x 2 – 4 Table of Values Graph y = x 2 – 4 for the values -3 < x < 3 (-3) 2 - 4 5 -3,5 (-2) 2 - 4 0 -2,0 (-1) 2 - 4 -3 -1,-3 (0) 2 - 4 -4 0,-4 (1) 2 - 4 -3 1,-3 (2) 2 - 4 0 2,0 (3) 2 - 4 5 3,5 -1,-3 (0,-4) (-2,0) (-3,5) (1,-3) (2,0) (3,5) y = x 2 - 4 Axis of symmetry- Turning point- x-intercepts- (0, -4) x = 0 Minimum (-2, 0)&(2, 0)

5. Course: Adv. Alg. & Trig. Aim: Graphing Parabola Graph y = -x 2 +2x + 5 for the values -2 < x < 4 Table of Values -(-2) 2 +2(-2)+5 -3 -2,-3 (-1,2) (0,5) (-2,-3) (4,-3) (3,2) (2,5) (1,6) Axis of symmetry- Turning point- x-intercepts- (1,6) x = 1 Maximum -(-1) 2 +2(-1)+5 2 -1,2 -(0) 2 +2(0)+5 5 0,5 -(1) 2 +2(1)+5 6 1,6 -(2) 2 +2(2)+5 5 2,5 -(3) 2 +2(3)+5 2 3,2 -(4) 2 +2(4)+5 -3 4,-3 (-?,0)&(?,0) y = -x 2 +2x + 5 How could we find the values of the x-intercepts or roots of y = -x 2 + 2x + 5?

6. Course: Adv. Alg. & Trig. Aim: Graphing Parabola A parabola is symmetrical about a line called the axis of symmetry. y = ax 2 + bx + c axis of symmetry turning point

7. Course: Adv. Alg. & Trig. Aim: Graphing Parabola Axis of symmetry of a parabola y = ax 2 + bx + c is the equation The x-value of the turning point equals -b/2a and the y-value can be found by substituting -b/2a for x in the equation y = ax 2 + bx + c. Axis of symmetry Axis of symmetry and the turning point Turning point of the parabola is always found on the axis of symmetry.

8. Course: Adv. Alg. & Trig. Aim: Graphing Parabola When a > 0 the parabola is a “valley” that opens upward. The curve falling until it reaches a lowest point, a minimum point. Then the curve turns and begins to rise (turning point). When a < 0 the parabola is a “hill” that opens downward. The curve is rising until it reaches a highest point, a maximum point. Then the curve turns and begins to fall (turning point). y = ax 2 + bx + c

9. Course: Adv. Alg. & Trig. Aim: Graphing Parabola The absolute value of a determines “fatness”. y = ax 2 + bx + c a = 1 a < 1 a > 1 As a increases, the shape of the parabola gets “thinner”. As a decreases in value, the shape of the parabola gets fatter or wider.

10. Course: Adv. Alg. & Trig. Aim: Graphing Parabola The value of c is the y-intercept of the parabola or the point where the parabola crosses the y-axis. y = ax 2 + bx + c c = +10 c = +5 c = 0 c = -6 Ex. y = 4.34x 2 - 15.445x + 1.456 y-intercept is +1.456

11. Course: Adv. Alg. & Trig. Aim: Graphing Parabola Model Problem Write an equation of the axis of symmetry of the graph of y = 3x 2 + 12x – 2 and find the coordinates of the turning point. 1) Establish what a, b and c are for the equation y = 3x 2 + 12x – 2 The equation for finding the axis of symmetry. 2) Evaluate for x = -b/2a a = 3b = 12c = -2 x = -12/2(3) = -12/6 = -2 3) To find turning point, evaluate y = 3x 2 + 12x – 2 when x = -2 Axis of symmetry is equation x = -2 y = 3(-2) 2 + 12(-2) – 2 = -14 coordinates of turning point - (-2, -14)

12. Course: Adv. Alg. & Trig. Aim: Graphing Parabola Model Problems Which is an equation of the graph shown? 1) y = x 2 – 4x + 4 2) y = x 2 + 4x + 4 3) y = -x 2 – 4x + 4 4) y = -x 2 + 4x + 4 Since the parabola opens upward eliminate 3) and 4). Find the axis of symmetry for equation 1) equation 2) x = -b/2a = -(-4)/2(1) = 2 x = -b/2a = -(4)/2(1) = -2 x = -2 Since a. of s. of shown graph is x = -2, choice 2) is correct answer.

13. Course: Adv. Alg. & Trig. Aim: Graphing Parabola Model Problem – How to Start Table of Values Graph y = x 2 - 4x + 3 (-1) 2 -4(-1)+3 8 -1,8 (0,3) (-1,8) (1,0) (2,-1) (3,0) (4,3) (1,8) x = 2 (0) 2 -4(0)+3 3 0,3 (1) 2 -4(1)+3 0 1,0 (2) 2 -4(2)+3 -1 2,-1 (3) 2 -4(3)+3 0 3,0 (4) 2 -4(4)+3 3 4,3 (5) 2 -4(5)+3 8 5,8 Since interval for table of values is not given, find the axis of symmetry and use 3 interval values to the left and 3 interval values to the right of the axis. Axis of symmetry - x = -(-4)/2(1) = 2 Values for table: -1, 0, 1, 2, 3, 4, 5

14. Course: Adv. Alg. & Trig. Aim: Graphing Parabola y = x 2 and x = y 2 What is the difference between y = x 2 and x = y 2 ? y = x 2 Describe the transformation that changes y = x 2 to x = y 2. x = y 2 Rotation of 90 0 about the origin.

15. Course: Adv. Alg. & Trig. Aim: Graphing Parabola Do Now: Aim: How do we graph a parabola? Write an equation of the axis of symmetry of the graph of y = 3x 2 + 12x – 2 and find the coordinates of the turning point.