Graphs of Quadratic Functions Part 2 Graph Quadratic

Graphs of Quadratic Functions Part 2

Graph Quadratic Functions Refresher: Standard Form: f(x) = 2 ax + Shape: Parabola When in standard form, If a is positive, the parabola opens up y = ax 2+bx+c If a is negative, the parabola opens down y = -ax 2+bx+c bx + c

Parts of Parabolas Refresher : Axis of Symmetry Vertex Find the equation for the axis of symmetry from points: 1. (3, 10) and (15, 10) 1. Since these ordered pairs are points of symmetry (same yvalue), you can find the axis of symmetry. 2. The middle of 3 and 15 is 9. 3. The axis of symmetry would be x = 9.

Parts of Parabolas Refresher : y = x 2 – 2 x – 3 y = (1)2 – 2(1) – 3 y = -4 Vertex (1, -4)

Graph Quadratic Functions Refresher:

Quadratic Graphs Vocabulary x-intercepts = Where the function crosses the x axis. There could be 2, 1, or no intercepts. Synonyms for x-intercepts: “roots of the function”, “zeros of the function”, “solutions to the function” Minimum value of function = y-value of minimum = the y coordinate of the vertex if parabola opens up and vertex is a min. Maximum Value of function = y-value of maximum= the y coordinate of the vertex if parabola opens down and if vertex is a max.

Transformations on Quadratic Graphs Translate = slide or shift adding/subtracting to the y value shifts up and down (it adds onto the end of the function and changes the y-int) c(x)=x 2 + 3 is 3 units higher than g(x)=x 2 d(x)=x 2 - 3 is 3 units lower than g(x)=x 2 adding/subtracting to the x value shifts right and left (it affects the zeros) k(x)=(x - 3)2 is 3 units to the right of g(x)=x 2 j(x)=(x + 3)2 is 3 units to the left of g(x)=x 2

Transformations on Quadratic Graphs Dilate = to change size / stretch

Quadratic Graphs Vocabulary End Behavior of a Graph: If a > 0, then f(x) opens up and has a minimum y value (at the coordinate of the vertex) f is decreasing for x-values less than, or to the left of, the vertex, f is increasing for x-values greater than, or to the right of, the vertex If a < 0, then f(x) opens down has a maximum y value (at x-coordinate of the vertex) f is increasing for x-values less than, or to the left of, the vertex, f is decreasing for x-values greater than, or to the right of, the vertex

Graphs of Quadratic Functions – Graph A Look at the table and the graph, state points of symmetry. x f(x) -1 8 0 3 (-1, 8) and (5, 8) 1 0 2 -1 3 0 4 3 5 8 (0, 3) and (4, 3) What interval is the graph increasing? [2, ∞] What interval is the graph decreasing? [-∞, 2] Domain of Graph? Set of all real numbers Range of Graph? Set of all real numbers > -1 y > -1

Graphs of Quadratic Functions – Graph B Look at the table and the graph, state points of symmetry. x f(x) -5 -5 -4 0 (-5, -5) and (1, -5) -3 3 (-4, 0) and (0, 0) -2 4 -1 3 0 0 1 -5 What interval is the graph increasing? [-∞, -2] What interval is the graph decreasing? [-2, ∞] Domain of Graph? Set of all real numbers Range of Graph? Set of all real numbers < 4 y<4

Patterns in the tables of values… Table A Table B x f(x) -1 8 -5 -5 0 3 -4 0 1 0 -3 3 2 -1 -2 4 3 0 -1 3 4 3 0 0 5 8 1 -5 Do Tables A and B … have the same y-intercepts? have the same x-intercepts? Have the same maximum / minimum? Both open up? Are both symmetric? What patterns do you see in the tables of values? How can we know the x-coordinate of the vertex by looking at two symmetric points? How can you tell if they will open up or down and have the same end behavior without graphing?