Name: Date: Topic: Solving & Graphing Quadratic Functions/Equations Essential Question: How can you solve quadratic equations? Warm-Up : Factor 1. 49p 2 – d 4 + 4d 3 – 6d 2 – 4d Solve for x: 3. 2x x + 6

Quadratic Function (y = ax 2 + bx + c) y = 3x 2 y = x y = x 2 – x – 2

Vocabulary: 1.Quadratic Parent Function 2.Parabola = the graph of a quadratic function is a U- shaped curved. 3.Axis of Symmetry – divide the graph into two halves Continue The line of symmetry ALWAYS passes through the vertex.

Vertex Minimum Vertex Maximum 4. Vertex Minimum – lowest point of the parabola Maximum – the highest point of the parabola.

y = x 2 a = 1, b = 0, c = 0 Minimum point (0,0) Axis of symmetry x=0 y=x 2 y = ax 2 + bx + c

Find the line of symmetry of y = 3x 2 – 18x + 7 Finding the Line of Symmetry When a quadratic function is in standard form The equation of the line of symmetry is y = ax 2 + bx + c, For example… Using the formula… Thus, the line of symmetry is x = 3. How do I find my vertex? Finding the Vertex What is the vertex?

Another example: 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)

STEP 1: Find the line of symmetry Let's Graph ONE! Try … y = 2x 2 – 4x – 1 A Quadratic Function in Standard Form STEP 2: Find the vertex STEP 4: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. STEP 3: Find the y-intercept. x2x 2 – 4x – 1y(x, y)

Step 5: Lets Graph it! y = 2x 2 – 4x – 1 A Quadratic Function in Standard Form

What happen if we change the value of a and c ? y=3x 2 y=-3x 2 y=4x 2 +3 y=-4x 2 -2

Conclusion to Quadratic Function (y = ax 2 +bx+c) When a is positive, When a is negative, When c is positive When c is negative the graph concaves downward. the graph concaves upward. the graph moves up. the graph moves down.

Solving Quadratic Equations Quadratic Formula x 2 – 2x – 8 = 0 Method #1: Hint: Quadratic equation, must equal 0

x 2 – 4x = 21 Example #2

 Factoring x 2 - 2x = 0 Factor in order to solve the equation. y=x 2 -2x Hints:  Remember to ask yourself does the function have a GCF.  Find the x intercept. Answer: Two solutions, x=0 and x=2. Method #2:

Page 538 (8, 10) Page 546 (43) Page 549 (a, b, c) Group Work: Independent Work: Group 1: #7, #20 Group 2: #8, #21 Group 5: #11, #24 Group 6: #12, #25 Page 544 – 545 Group 3: #9, #22 Group 4: #10, #23 Group 7: #13, #28 Group 8: #14, #30

Solve the following equations by factoring: 1.x 2 = 25 2.x 2 – 8 = - 7x 3.x 2 – 12x = x 2 = 7x 5. m 2 – 3m = r – 8 = - r x 2 = x 2 = 9x – 20

y = -x 2 + 2x + 8

Find the Solutions y=x 2 -4 y=x 2 +2x-15 y=-x 2 +5 y=-x 2 -1

Find the solutions y=x 2 +2x+1 y=-x 2 +4x-1

HLA#5: Page 538 (7, 8) Page 544 (2, 3, 4)