1. Ch. 4 Pre-test 1.Graph the function : y = – 4x 2 Then label the vertex and axis of symmetry. 2.Write the quadratic function in standard form : y = (x – 5)(x + 3) 3. Factor the expression : x 2 – 7x – 18 4. Solve the equation : 11s 2 – 44 = 0 5. Simplify the expression :

2. 4.1 Graph Quadratic Functions in Standard Form Key Vocabulary: -standard form of a quadratic function: y = ax 2 + bx + c where a ≠ 0. -The graph is a parabola. -The lowest or highest point is the vertex: (x, y) -The axis of symmetry divides the parabola in symmetric halves.

3. Graph a quadratic function: 2. Make a Table: xy -24 1 00 11 24 3. Plot the points on the graph. 4. Draw a smooth curve through the points. Forms a parabola. 5.Identify the axis of symmetry. 6.Identify the vertex. 1. Identify a, b, c constants. y = ax 2 + bx + c

4. Key Concept Sdf asdf Sadf Asdf asdf The axis of symmetry on the parent graph is x = 0. The Vertex sdf The vertex is (0, 0). Sa asd The parent function is y = x 2.

5. Graph y = ax 2 + bx + c. 2. Make a Table: xy -24 1 00 11 24 3. Plot the points on the graph. 4. Draw a smooth curve through the points. Forms a parabola. 5.Identify the axis of symmetry. 6.Identify the vertex. 1. Identify a, b, c constants.

6. EXAMPLE 1 Graph y = 2x 2. Compare the graph with y = x 2. Notice that both graphs have the same axis of symmetry ( x = 0) and the same vertex at (0,0). Compare the two graphs: Graph y = –. Compare with the x 2 + 3 1212 graph of y = x 2. Notice that both graphs have the same axis of symmetry x = 0. But different vertex points. One at (0, 0) and the other at (0, 3).

7. Graph y = ax 2 + bx + c. 2. Make a Table: xy -24 1 00 11 24 3. Plot the points on the graph. 4. Draw a smooth curve through the points. Forms a parabola. 5.Identify the axis of symmetry. 6.Identify the vertex. 1. Identify a, b, c constants.

8. GUIDED PRACTICEGraph the function. Compare the graph with the graph of y = x 2. 1. y = – 4x 2 Compared with y = x 2 Same axis of symmetry and vertex, opens down, and is narrower. 2. y = – x 2 – 5 Same axis of symmetry, vertex is shifted down 5 units, and opens down.

9. So far, no b term (b = 0)... now we add the b term: Properties of the graph of y = ax 2 + bx + c Characteristics of the graph of y = ax 2 + bx + c: The graph opens up if a > 0 (a is a positive #) and opens down if a < 0 (a is a negative #). The vertex has x-coordinate x =. Substitute to find y-coordinate. The axis of symmetry is x =. The y-intercept is c, so the point ( 0, c ) is on the parabola. As df’

10. Graph y = ax 2 + bx + c. 1. Identify a, b, c constants. 2. Find the vertex: x = Substitute x into function for y-coordinate. Form for vertex point: (x, y) 3. Identify the axis of symmetry: x-coordinate of vertex. 4. Graph the vertex point and axis of symmetry. 5. Make a table. (Use points on both sides of vertex.) 6. Graph the points from the table and connect the points with a smooth curve forming a parabola.

11. EXAMPLE 3 Graph y = 2x 2 – 8x + 6. Label the vertex and axis of symmetry. 1. Find the vertex: x = b2a b2a = (– 8) 2(2) –– = 2 Now find the y - coordinate of the vertex: y = 2(2) 2 – 8(2) + 6 = – 2 So, the vertex is (2, – 2). 2. The axis of symmetry is: x = 2. a = 2 b = -8 c = 62 > 0 so parabola opens up 3. Make a table: xy 16 06 12341234 0 -2 0 6 4. Graph:

12. Graph y = ax 2 + bx + c. 1. Identify a, b, c constants. 2. Find the vertex: x = Substitute x into function for y-coordinate. Form for vertex point: (x, y) 3. Identify the axis of symmetry: x-coordinate of vertex. 4. Graph the vertex point and axis of symmetry. 5. Make a table. (Use points on both sides of vertex.) 6. Graph the points from the table and connect the points with a smooth curve forming a parabola.

13. GUIDED PRACTICE Graph the function. Label the vertex and axis of symmetry. 4. y = x 2 – 2x – 1 The vertex is (1, – 2). The axis of symmetry x = 1.

14. Key Concept Minimum and Maximum Values Lindsey Anderson

15. To find the minimum or maximum value of a quadratic function: 1.Is a > 0 or is a < 0. (i.e. is a negative or positive?) a is negative – maximum a is positive – minimum 2.Find the vertex: (substitute for y) 3.The max. or min. value is the y-coordinate of the vertex. x = – b2a b2a

16. EXAMPLE 4 Tell whether the function y = 3x 2 – 18x + 20 has a minimum value or a maximum value. Then find the minimum or maximum value. a > 0 (a is positive), so the function has a minimum value. x = – b2a b2a = – (– 18) 2(3) = 3 y = 3(3) 2 – 18(3) + 20 = –7 The minimum value is y = –7 at the point (3, -7). Homework 4.1: p. 240: 3-52 (EOP) a = 3 b = -18 c = 20 Find the vertex: