Analyze Graphs of Quadratic Functions

Quadratic Graphs Vocabulary End Behavior of a Graph: Describes if a function opens up or down and will have a min. or max. If a is positive (a > 0), then f(x) opens up has a minimum value (at the y-coordinate of the vertex, like range) If a is negative (a < 0), then f(x) opens down has a maximum value (at the y-coordinate of the vertex, like range) To find the min/max value, find vertex using . The y-coordinate of vertex is the function’s min/max value

Quadratic Graphs Vocabulary x-intercepts = Where the function crosses the x axis. There could be 2, 1, or no intercepts. The x-intercepts can also be called “roots of the function”, “zeros of the function”, “solutions to the function” To find the x-intercepts, solve the quadratic. The x-intercepts are the solutions Solve by factoring or any other method you like

Find the x-intercepts / roots and the min or max value Example 2: y = x2 - 2x - 3 Example 1: y = x2 + 8x Find the x-intercept/roots by solving the quadratic y=(x)(x+8) 0=(x)(x+8) x=0 x=-8 x-intercepts / roots: (0,0) and (0,-8) y=(x-3)(x+1) 0=(x-3)(x+1) x=3 x=-1 x-intercepts / roots: (0,3) and (0,-1) Find the vertex in order to identify the min/max of the function x= −𝑏 2𝑎 = −8 2(1) = -4 y=(-4)2 + 4(-4) y=-16 vertex (-4,-16) Min. value is -16 x= −𝑏 2𝑎 = −(−2) 2(1) = 1 y=(1)2 - 2(1) - 3 y=-4 vertex (1,-4) Min. value is -4

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)=x2 + 3 is 3 units higher than g(x)=x2 d(x)=x2 - 3 is 3 units lower than g(x)=x2 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)=x2 j(x)=(x + 3)2 is 3 units to the left of g(x)=x2

Transformations on Quadratic Graphs Dilate = to change size / stretch Multiplying by a number > 1 creates a vertical stretch Makes the parabola more narrow Changes the a value in standard form c(x)=2x2 is more narrow than g(x)=x2 (vertically stretched) Multiplying by a number between 0 and 1 (a fraction less than 1) creates a horizontal stretch Makes the parabola wider c(x)= 𝟏 𝟐 x2 is wider than g(x)=x2 (horizontally stretched) Multiplying by a negative flips the parabola to open down (changes sign of a value)

Patterns in the tables of values… Table A Table B 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? x f(x) -1 8 3 1 2 4 5 x f(x) -5 -4 -3 3 -2 4 -1 1 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?