4.1 Notes – Graph Quadratic Functions in Standard Form

Parabola that is a function in the shape of a U Quadratic Function Standard Form Vertex Axis of Symmetry Parabola that is a function in the shape of a U y = ax2 + bx + c Max or Min of a parabola Line that splits the graph vertically

Transformations of a, b, and c. y = ax2 + bx + c Positive # Negative # a Opens up Opens down Has minimum Has maximum a = 1, normal a = –1, normal Fraction, wide Fraction, wide a > 1, skinny a < –1, skinny

Set y = 0, factor the equation and solve for x x – intercepts Set y = 0, factor the equation and solve for x Two answers One answer No Real answers

Shift vertex left or right y = ax2 + bx + c b Shift vertex left or right c y – intercept

3. Sketch the graph of the quadratic equations 3. Sketch the graph of the quadratic equations. Identify the vertex, axis of symmetry, the minimum or maximum, the x­-intercepts, and the y-intercepts. This is the parent function.

3. Sketch the graph of the quadratic equations 3. Sketch the graph of the quadratic equations. Identify the vertex, axis of symmetry, the minimum or maximum, the x­-intercepts, and the y-intercepts. x (x,y) 2 (2, 4) 1 (1,1) (0,0) –1 (–1,1) –2 (–2, 4)

(0,0) x = 0 y = 0 Vertex: ________________ axis of symmetry: ________ circle one: min or max min/max value: _________ x – intercept(s): _________ y-intercept: _____________ x = 0 y = 0

(0,0) Vertex: ________________ axis of symmetry: ________ circle one: min or max min/max value: _________ x – intercept(s): _________ y-intercept: _____________ x = 0 y = 0 (2, 4) (1,1) (0,0) (–1,1) (–2, 4)

1. Put the equation in standard from 1. Put the equation in standard from. Determine if the graph opens up or down and if it is normal, skinny, or wide. Standard Form: Opens up wide

1. Put the equation in standard from 1. Put the equation in standard from. Determine if the graph opens up or down and if it is normal, skinny, or wide. Standard Form: Opens down skinny

1. Put the equation in standard from 1. Put the equation in standard from. Determine if the graph opens up or down and if it is normal, skinny, or wide. Standard Form: Opens down normal

2. Identify the vertex. State the max or min. (2, –13) Vertex: ________________ min/max value: _________ y = –13

2. Identify the vertex. State the max or min. (1, 11) Vertex: ________________ min/max value: _________ y = 11

2. Identify the vertex. State the max or min. (0, 5) Vertex: ________________ min/max value: _________ y = 5

3. Sketch the graph of the quadratic equations 3. Sketch the graph of the quadratic equations. Identify the vertex, axis of symmetry, the minimum or maximum, the x­-intercepts, and the y-intercepts.

3. Sketch the graph of the quadratic equations 3. Sketch the graph of the quadratic equations. Identify the vertex, axis of symmetry, the minimum or maximum, the x­-intercepts, and the y-intercepts. x (x,y) 2 (2, 5) 1 (1, 8) (0, 9) –1 (–1, 8) –2 (–2, 5)

(0, 9) x = 0 y = 9 3 x + 3 = 0 or x - 3 = 0 9 x = -3 x = 3 Vertex: ________________ axis of symmetry: ________ circle one: min or max min/max value: _________ x – intercept(s): _________ y-intercept: _____________ x = 0 y = 9 3 x + 3 = 0 or x - 3 = 0 9 x = -3 x = 3

(0, 9) Vertex: ________________ axis of symmetry: ________ circle one: min or max min/max value: _________ x – intercept(s): _________ y-intercept: _____________ x = 0 y = 9 3 9 (2, 5) (1, 8) (0, 9) (–1, 8) (–2, 5)

3. Sketch the graph of the quadratic equations 3. Sketch the graph of the quadratic equations. Identify the vertex, axis of symmetry, the minimum or maximum, the x­-intercepts, and the y-intercepts.

3. Sketch the graph of the quadratic equations 3. Sketch the graph of the quadratic equations. Identify the vertex, axis of symmetry, the minimum or maximum, the x­-intercepts, and the y-intercepts. x (x,y) 4 (4, 6) 2 (2, 0) (0, –2) –2 (–2, 0) –4 (–4, 6)

(0, –2) x = 0 y = –2  2 –2 Vertex: ________________ axis of symmetry: ________ circle one: min or max min/max value: _________ x – intercept(s): _________ y-intercept: _____________ x = 0 y = –2  2 –2

(0, –2) x = 0 y = –2  2 –2 (4, 6) (2, 0) (0, –2) (–2, 0) (–4, 6) Vertex: ________________ axis of symmetry: ________ circle one: min or max min/max value: _________ x – intercept(s): _________ y-intercept: _____________ x = 0 y = –2  2 –2 (4, 6) (2, 0) (0, –2) (–2, 0) (–4, 6)

3. Sketch the graph of the quadratic equations 3. Sketch the graph of the quadratic equations. Identify the vertex, axis of symmetry, the minimum or maximum, the x­-intercepts, and the y-intercepts.

3. Sketch the graph of the quadratic equations 3. Sketch the graph of the quadratic equations. Identify the vertex, axis of symmetry, the minimum or maximum, the x­-intercepts, and the y-intercepts. x (x,y) 1 (1, –9) (0, 0) (–1, 3) –1 –2 (–2, 0) –3 (–3, –9)

(–1, 3) x = –1 -3x -3x -3x(x + 2) = 0 y = 3 -3x = 0 or x + 2 = 0 0, -2 Vertex: ________________ axis of symmetry: ________ circle one: min or max min/max value: _________ x – intercept(s): _________ y-intercept: _____________ x = –1 -3x -3x -3x(x + 2) = 0 y = 3 -3x = 0 or x + 2 = 0 0, -2 x = 0 x = -2

(–1, 3) x = –1 y = 3 0, -2 (1, –9) (0, 0) (–1, 3) (–2, 0) (–3, –9) Vertex: ________________ axis of symmetry: ________ circle one: min or max min/max value: _________ x – intercept(s): _________ y-intercept: _____________ x = –1 y = 3 0, -2 (1, –9) (0, 0) (–1, 3) (–2, 0) (–3, –9)

3. Sketch the graph of the quadratic equations 3. Sketch the graph of the quadratic equations. Identify the vertex, axis of symmetry, the minimum or maximum, the x­-intercepts, and the y-intercepts.

3. Sketch the graph of the quadratic equations 3. Sketch the graph of the quadratic equations. Identify the vertex, axis of symmetry, the minimum or maximum, the x­-intercepts, and the y-intercepts. x (x,y) (0, 3) –1 (–1, 0) (–2, –1) –2 –3 (–3, 0) –4 (–4, 3)

(–2, –1) x = -2 x 3 x 1 y = –1 3x + x -1, -3 (x + 3)(x + 1) = 0 3 Vertex: ________________ axis of symmetry: ________ circle one: min or max min/max value: _________ x – intercept(s): _________ y-intercept: _____________ x = -2 x 3 x 1 y = –1 3x + x -1, -3 (x + 3)(x + 1) = 0 3 x + 3 = 0 or x + 1 = 0 x = –3 x = –1

(–2, –1) x = -2 y = –1 -1, -3 3 (0, 3) (–1, 0) (–2, –1) (–3, 0) Vertex: ________________ axis of symmetry: ________ circle one: min or max min/max value: _________ x – intercept(s): _________ y-intercept: _____________ x = -2 y = –1 -1, -3 3 (0, 3) (–1, 0) (–2, –1) (–3, 0) (–4, 3)

3. Sketch the graph of the quadratic equations 3. Sketch the graph of the quadratic equations. Identify the vertex, axis of symmetry, the minimum or maximum, the x­-intercepts, and the y-intercepts.

3. Sketch the graph of the quadratic equations 3. Sketch the graph of the quadratic equations. Identify the vertex, axis of symmetry, the minimum or maximum, the x­-intercepts, and the y-intercepts. x (x,y) 3 (3, –8) 2 (2, –2) (1, 0) 1 (0, –2) –1 (–1, –8)

(1, 0) -2 -2 -2 x = 1 y = 0 1 -2 -2(x2 – 2x + 1) = 0 x -1 x -1 -x + -x Vertex: ________________ axis of symmetry: ________ circle one: min or max min/max value: _________ x – intercept(s): _________ y-intercept: _____________ -2 -2 -2 x = 1 -2(x2 – 2x + 1) = 0 x -1 y = 0 x -1 1 -x + -x -2(x – 1)2 = 0 -2 (x – 1)2 = 0 x – 1 = 0 x = 1

(1, 0) x = 1 y = 0 1 -2 (3, –8) (2, –2) (1, 0) (0, –2) (–1, –8) Vertex: ________________ axis of symmetry: ________ circle one: min or max min/max value: _________ x – intercept(s): _________ y-intercept: _____________ x = 1 y = 0 1 -2 (3, –8) (2, –2) (1, 0) (0, –2) (–1, –8)