Graphs of Quadratic Functions Part 1 UPDATE!!! Graphs of Quadratic Functions Part 1

Graph Quadratic Functions: Standard Form: y = ax2 + bx + c Shape: Parabola When in standard form, If a is positive, the parabola opens up y = ax2+bx+c If a is negative, the parabola opens down y = -ax2+bx+c

Will It Open Up or Down? y = 4x2 + 7x – 4 y = -6.5x2 + 9 *must be in standard form

Parts of Parabolas: Axis of Symmetry: Vertex: Highest or lowest point of the graph (the max or min of the function) Lies on the axis of symmetry Axis of Symmetry: Line of symmetry that divides parabola into two parts that are mirror image of each other. Cuts through the vertex Axis of Symmetry Vertex

The axis of symmetry is the vertical line passing through y = ax2+bx+c Find Vertex and Write the equation of the axis of symmetry for y = 3x2+8x-6 1st: Identify a, b, and c 2nd: Plug into a=3, b=8, c=-6 Axis of symmetry is vertical line x= -4 3

Find the vertex of y = 3x2+8x-6 Continued… 3rd: Plug x value into function to solve for y y = 3( −4 3 )2 + 8 ( −4 3 ) - 6 The vertex has an x-coordinate of −𝑏 2𝑎 Because it lies on the axis of symmetry y = 3( 16 9 ) + 8 ( −4 3 ) - 6 y = 16 3 - 32 3 - 6 y = -11 1 3 Vertex is point (-1 𝟏 𝟑 , -11 𝟏 𝟑 )

x=0 is the axis of symmetry Find the Vertex and Graph: y = x2 First find the vertex. y = ax2+bx+c a=1, b=0, and c=0 x=0 is the axis of symmetry This is also the x value of the vertex, now find the y value. If x = 0, y = (0)2 Vertex = (0,0)

Since the vertex is (0,0), pick an + and a - x value Example: y = x2 Make a table for y = x2 Once you find the vertex, pick at least 2 more points to see how to graph the parabola, plot them and their mirror images to make it symmetrical Since the vertex is (0,0), pick an + and a - x value Once you find the vertex, pick a point to the left and one to the right to see how to graph the parabola.

Graphing Quadratic Functions UPDATE!!! 5 Simple Steps (make sure to identify the a, b, and c values) Find the equation of the axis of symmetry & draw it on the graph Find the vertex coordinates & plot vertex on graph Find and plot the y-intercept point Find and plot another point on the other side of the vertex Sketch the curve and reflect it across the  axis of symmetry

A is positive 1, so the parabola opens up with (0,0) as the low point. Graph Points Line of symmetry A is positive 1, so the parabola opens up with (0,0) as the low point.

GRAPH: y = x2-x-6 Identify the a, b, and c values First find the vertex Make a table with an x value to the right and left of the vertex x value Graph these points and connect. Label the vertex

Find vertex and plug in to find y. value to have high or low point. 1.

The x value of the vertex is 1/2 Now find the y value of the vertex by plugging x back into the equation. y = x2-x-6 y = (1/2)2 – ½ - 6 The y value is -25/4. Now pick a point to the left and right of ½.

y = x2-x-6 GRAPH: I try to pick points equal distance from the vertex x value. I also tried 0 here.

Y=x2-x-6 Vertex low line of symmetry x = opens up (a positive) If you pick points equal distance from the vertex, you will get the same value for y. Your graph will be symmetrical. line of symmetry x = opens up (a positive)

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

2 - - 2 - 2 - 2

y=-2x2+2x+1