Graphs of Quadratic Functions Day 2

The axis of symmetry is the vertical line passing through the vertex 1. Find the equation for the axis of symmetry just from symmetric points: (3, 10) and (15, 10) Since these ordered pairs are points of symmetry (same y-value, height on graph), you can find the axis of symmetry by finding the middle. The middle of 3 and 15 is 9. The axis of symmetry would be x = 9. 2. Find the equation for the axis of symmetry from formula: y = ax2+bx+c

x = -1 is the axis of symmetry Graph y = - 𝟏 𝟐 x2 - x + 4 1. Find the axis of symmetry. y = ax2+bx+c a=- 𝟏 𝟐 , b=-1, and c=4 x = −𝑏 2𝑎 = −(−1) 2(−1/2) = 1 −1 = -1 x = -1 is the axis of symmetry 2. Find the vertex. plug in x = -1 y = - 𝟏 𝟐 (-1)2 – (-1) + 4 y=4.5 Vertex = (-1,4.5)

y = ax2+bx+c a=- 𝟏 𝟐 , b=-1, and c=4 Graph y = - 𝟏 𝟐 x2 - x + 4 continued 3. Graph 2 more points Try the y-intercept (c value) y = ax2+bx+c a=- 𝟏 𝟐 , b=-1, and c=4 y-intercept (0,4) Find more points Select any x value you want and plug into function to find y value (make easy choices, ie. whole numbers Select x=1, and plug in y = - 𝟏 𝟐 (1)2 – (1) + 4 y = - 𝟏 𝟐 - 1 + 4 y=2.5 makes point (1,2.5)

Graph y = - 𝟏 𝟐 x2 - x + 4 continued 4. Graph all points and mirror images to make symmetric parabola axis of symmetry x=-1 Vertex (-1,4.5) (0,4) and mirror image (-2,4) (1,2.5) and mirror image (-3,2.5) Check: Opens down because a is negative

Graph Quadratic Example Notes:

x = 𝟏 𝟐 is the axis of symmetry Graph y = x2-x-6 1. Find the axis of symmetry. y = ax2+bx+c a=1, b=-1, and c=-6 x = −𝑏 2𝑎 = −(−1) 2(1) = 1 2 x = 𝟏 𝟐 is the axis of symmetry 2. Find the vertex. plug in x = 𝟏 𝟐 y = ( 𝟏 𝟐 )2 – ( 𝟏 𝟐 ) - 6 y = -6 𝟏 𝟒 Vertex = ( 𝟏 𝟐 , -6 𝟏 𝟒 )

y = ax2+bx+c a=1, b=-1, and c=-6 Graph y = x2-x-6 continued 3. Graph at least 2 more points Try the y-intercept (c value) y = ax2+bx+c a=1, b=-1, and c=-6 y-intercept (0,-6) Find more points Select any x value you want and plug into function to find y value (make easy choices, ie. whole numbers Select x=2, and plug in Select x=3, and plug in y = (𝟐)2 – (𝟐) - 6 y=-4 makes point (2,-4) y = (𝟑)2 – (𝟑) - 6 y=0 makes point (3,0)

Graph y = x2-x-6 continued 4. Graph all points and mirror images to make symmetric parabola axis of symmetry x = 𝟏 𝟐 Vertex ( 𝟏 𝟐 , -6 𝟏 𝟒 ) (0,-6) and mirror image (1,-6) (2,-4) and mirror image (-1,-4) (3,0) and mirror image (-2,0) Check: Opens up because a is positive

Additional Practice Problems if needed

a is negative: opens down Graph: y= -2x2+2x+1 a is negative: opens down Find Line of Symmetry = 1 2 Find the y value, then pick more points to see how to draw the parabola.

2 - - 2 - 2 - 2

y=-2x2+2x+1

Graph: y= x2+5x-14 Plug back in for y and solve Will open up b/c a is positive Find axis of symmetry and the vertex Plug back in for y and solve

Change to common denominators Vertex is

Vertex is Can also find x intercepts of y = x2 + 5x - 14 by factoring to find solutions (2,0) and (-7,0) Vertex is y -7 x 2