1. Graphing Quadratics

2. Finding the Vertex We know the line of symmetry always goes through the vertex. Thus, the line of symmetry gives us the x – coordinate of the vertex. To find the y – coordinate of the vertex, we need to plug the x – value into the original equation. STEP 1: Find the line of symmetry STEP 2: Plug the x – value into the original equation to find the y value. y = –2x 2 + 8x –3 y = –2(2) 2 + 8(2) –3 y = –2(4)+ 8(2) –3 y = –8+ 16 –3 y = 5 Therefore, the vertex is (2, 5)

3. A Quadratic Function in Standard Form The standard form of a quadratic function is given by y = ax 2 + bx + c There are 3 steps to graphing a parabola in standard form. STEP 1: Find the line of symmetry STEP 2: Find the vertex STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. Plug in the line of symmetry (x – value) to obtain the y – value of the vertex. MAKE A TABLE using x – values close to the line of symmetry. USE the equation

4. STEP 1: Find the line of symmetry Let's Graph ONE! Try … y = 2x 2 – 4x – 1 A Quadratic Function in Standard Form Thus the line of symmetry is x = 1

5. Let's Graph ONE! Try … y = 2x 2 – 4x – 1 STEP 2: Find the vertex A Quadratic Function in Standard Form Thus the vertex is (1,–3). Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex.

6. 5 –1 Let's Graph ONE! Try … y = 2x 2 – 4x – 1 STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. A Quadratic Function in Standard Form 3 2 yx

7. 1) Graphing Parabolas Example 2: Graph y = x 2 – 2x – 3.

8. 1) Graphing Parabolas Example 2: Graph y = x 2 – 2x – 3. Step 1: Find the axis of symmetry. Axis of symmetry: Equation for the axis of symmetry: x = 1

9. 1) Graphing Parabolas Example 2: Graph y = x 2 – 2x – 3. Step 2: Find the vertex. Sub x = 1 into the equation, solve for y. y = (1) 2 – 2(1) – 3 y = -4 The vertex is at (1, -4).

10. 1) Graphing Parabolas Example 2: Graph y = x 2 – 2x – 3. Step 3: Plot points and sketch the graph. xy 1-4 2 3 4

11. 1) Graphing Parabolas Example 2: Graph y = x 2 – 2x – 3. Step 3: Plot points and sketch the graph. xy 1-4 2(2) 2 – 2(2) – 3 = -3 3(3) 2 – 2(3) – 3 = 0 4(4) 2 – 2(4) – 3 = 5

12. 1) Graphing Parabolas Example 2: Graph y = x 2 – 2x – 3.

13. EXAMPLE Graph y = 2x 2 -8x +6 Solution: The coefficients for this function are a = 2, b = -8, c = 6. Since a>0, the parabola opens up. The x-coordinate is: x = -b/2a, x = -(-8)/2(2) x = 2 The y-coordinate is: y = 2(2) 2 -8(2)+6 y = -2 Hence, the vertex is (2,-2).

14. EXAMPLE(contd.)  Draw the vertex (2,-2) on graph.  Draw the axis of symmetry x=-b/2a.  Draw two points on one side of the axis of symmetry such as (1,0) and (0,6).  Use symmetry to plot two more points such as (3,0), (4,6).  Draw parabola through the plotted points. (2,-2) (1,0) (0,6) (3,0) (4,6) Axis of symmetry x y

15. VERTEX FORM OF QUADRATIC EQUATION y = a(x - h) 2 + k  The vertex is (h,k).  The axis of symmetry is x = h.

16. GRAPHING A QUADRATIC FUNCTION IN VERTEX FORM (-3,4) (-7,-4) (-1,2) (-5,2) (1,-4) Axis of symmetry x y  Example y = -1/2(x + 3) 2 + 4 where a = -1/2, h = -3, k = 4. Since a<0 the parabola opens down.  To graph a function, first plot the vertex (h,k) = (-3,4).  Draw the axis of symmetry x = -3  Plot two points on one side of it, such as (- 1,2) and (1,-4).  Use the symmetry to complete the graph.

17. Your Turn! Analyze and Graph: y = (x + 4) 2 - 3. (-4,-3)

18. 18 Example: Basketball Example: A basketball is thrown from the free throw line from a height of six feet. What is the maximum height of the ball if the path of the ball is: The path is a parabola opening downward. The maximum height occurs at the vertex. At the vertex, So, the vertex is (9, 15). The maximum height of the ball is 15 feet.