Ch 9: Quadratic Equations C) Graphing Parabolas Objective: To graph quadratic functions.

Definitions Quadratic Function A quadratic function is a polynomial of the form y=ax2 + bx + c where “a” is unequal to 0. Note: “x” must have an exponent of 2 Parabola The graph of a quadratic function - generally “U” shaped. A “positive” parabola (when a > 0) opens up toward the sky. A “negative” parabola (when a < 0) opens down toward the ground. Vertex x = -b/(2a) The vertex is also referred to as “the turning point” of the parabola. It is the lowest point for a positive parabola and the highest point for a negative parabola. Axis of Symmetry A line drawn through the vertex that cuts the parabola into two equal sections.

Graphing the vertex Determine the values for a, b, and c. Solve for x by plugging “a” and “b” into the vertex formula: Plug the “x” value into quadratic function (y= ax2 + bx + c) and solve for “y” Plot the x,y coordinate on the graph

Graphing a Parabola Graph the coordinate of the vertex Choose any “x” value to the left of the vertex and solve for “y” Choose any “x” value to the right of the vertex and solve for “y” Hint: Choose “x = 0” for one of the values and then choose an x value equal distance from the vertex. 4) Plot the x,y coordinates from steps 2 & 3 5) Draw a curve through all three coordinates.

Important Fact! You need at least these 3 points to graph a quadratic function. The vertex A coordinate to the LEFT of the vertex A coordinate to the RIGHT of the vertex. Note: You must have at least one point to the right of the vertex and one point to the left of the vertex in order to graph a parabola.

Step 1: Finding a, b, c Equation must be in the form: 0 = ax2 + bx + c “a” is the coefficient for x2 (the number before x2) “b” is the coefficient for x (the number before x) “c” is the number without a variable Note: the sign (+ , −) is part of the number!

Example 1 Example 2 0 = 2x2 + 3x + 4 0 = 3x2 − 4x + 5 Example 3 b = 3 c = 4 a = 3 b = −4 c = 5 Example 3 Example 4 0 = x2 + 4x − 1 0 = −x2 + 2x − 3 a = 1 b = 4 c = −1 a = −1 b = 2 c = −3 Example 5 Example 6 0 = x2 − 3x 0 = −4x2 a = 1 b = −3 c = 0 a = −4 b = 0 c = 0

Classwork y = x2 − 4x + 3 y = x2 + 6x + 5 1) 2) a = 1 b = −4 c = 3

Classwork y = -x2 y = -4x2 3) 4) a = −1 b = 0 c = 0 a = −4 b = 0 c = 0

Step 2: Finding the Axis of Symmetry −b 2a Use the the formula: x = Example 1 Example 2 y = x2 + 4x + 4 y = −2x2 − 4x + 5 a = 1 b = 4 c = 4 a = −2 b = −4 c = 5 −b 2a −( ) 2( ) −b 2a −( ) 2( ) +4 −4 4 −4 2 −4 x = = −2 x = = = = = = −1 1 −2

Classwork y = x2 − 4x + 3 y = x2 + 6x + 5 −b 2a −( ) 2( ) +4 2 x = = 2 1) 2) a = 1 b = −4 c = 3 a = 1 b = 6 c = 5 −b 2a −( ) 2( ) −4 +4 2 x = = 2 −b 2a −( ) 2( ) −6 2 = = 6 x = = = = −3 1 1

Classwork y = -x2 y = -4x2 −b 2a −( ) 2( ) −2 x = = −b 2a −( ) 2( ) −8 3) 4) a = −1 b = 0 c = 0 a = −4 b = 0 c = 0 −b 2a −( ) 2( ) −2 x = = −b 2a −( ) 2( ) −8 = = x = = = = −1 −4

Step 3: Find the vertex Plug the “x” value into the equation and solve for “y” y = a(x)2 + b(x) + c Then plot the coordinate (x,y) on the graph Example 1 Example 2 y = x2 + 4x + 4 y = −2x2 − 4x + 5 a = 1 b = 4 c = 4 a = −2 b = −4 c = 5 −b 2a −( ) 2( ) −b 2a −( ) 2( ) +4 −4 4 −4 2 −4 x = = x = = = = = = −2 −1 1 −2 y = ( )2 + 4( ) + 4 y = −2( )2 + 4( ) + 4 −2 −2 1 1 = = 4 = = 6 −8 +4 −2 +4 +4 Plot coordinate (-2, 0) Plot coordinate (−1, 6)

Classwork y = x2 − 4x + 3 y = x2 + 6x + 5 −b 2a −( ) 2( ) −b 2a −( ) 1) 2) a = 1 b = −4 c = 3 a = 1 b = 6 c = 5 −b 2a −( ) 2( ) −b 2a −( ) 2( ) −6 2 −4 +4 2 6 x = = 2 x = = = = = = −3 1 1 y = ( )2 − 4( ) + 3 y = ( )2 + 6( ) + 5 −4 2 2 = −3 −3 =

Classwork y = -x2 y = -4x2 −b 2a −( ) 2( ) −b 2a −( ) 2( ) −8 −2 x = = 3) 4) a = −1 b = 0 c = 0 a = −4 b = 0 c = 0 −b 2a −( ) 2( ) −b 2a −( ) 2( ) −8 −2 x = = x = = = = = = −1 −4 y = −( )2 y = −4( )2 = =

Step 4: Finding a second point Choose a value for “x” and Plug it into the equation y = a(x)2 + b(x) + c Then plot the coordinate (x,y) on the graph Example 1 Example 2 y = x2 + 4x + 4 y = −2x2 − 4x + 5 I choose x = 0 I choose x = 0 y = ( )2 + 4( ) + 4 = 4 y = −2( )2 − 4( ) + 5 = 5

Classwork y = x2 − 4x + 3 y = x2 + 6x + 5 y = ( )2 − 4( ) + 3 = 3 1) 2) I choose x = 0 I choose x = 0 y = ( )2 − 4( ) + 3 = 3 y = ( )2 + 6( ) + 5 = 5

Classwork y = -x2 y = -4x2 y = −( )2 = −1 y = −4( )2 = −4 3) 4) I choose x = 1 I choose x = 1 y = −( )2 1 = −1 y = −4( )2 1 = −4

Step 5: Plotting the third point Fold your paper so the crease is on the Axis of Symmetry Plot the “mirror image” of the coordinate on the graph Draw a curve through all three points Example 1 Example 2 y = x2 + 4x + 4 y = −2x2 − 4x + 5

Classwork y = x2 − 4x + 3 y = x2 + 6x + 5 1) 2)

Classwork y = -x2 y = -4x2 3) 4)

Vertex: x = = = = 3 Axis of Symmetry x = 3 x y y = (3)2 - 6(3) + 2 Example 3: Graph y = x2 – 6x + 2 -b 2a -(-6) 2 (1) 6 2 Vertex: x = = = = 3 Axis of Symmetry x = 3 x y y = (3)2 - 6(3) + 2 = -7 x y Left Vertex 3 -7 Right Vertex (3,-7)

Draw a curve through the points Example 3: Graph y = x2 – 6x + 2 second point: Choose x = 0 Axis of Symmetry x = 3 y = (0)2 – 6(0) + 2 = 2 x y Right of Vertex Left of Vertex (6, 2) (0, 2) x y Left 2 Vertex 3 -7 Right 6 2 Vertex (3,-7) Draw a curve through the points

Vertex: x = = = = 0 Axis of Symmetry x = 0 x y y = (0)2 – 5 = -5 x y Example 4: Graph y = x2 - 5 -b 2a -(0) 2(1) 2 Vertex: x = = = = 0 Axis of Symmetry x = 0 x y y = (0)2 – 5 = -5 x y Left Vertex -5 Right Vertex (0,-5)

Draw a curve through the points Example 4: Graph y = x2 - 5 Second point: Choose x = 3 Axis of Symmetry y = (3)2 – 5 = 4 x = 0 x y Left of Vertex Right of Vertex (-3, 4) (3, 4) x y Left Vertex -5 Right 3 4 -3 -4 Vertex (0,-5) Draw a curve through the points

Vertex: x = = = = -1 Axis of Symmetry x = -1 x y Example 5: Graph y = -x2 - 2x + 4 -b 2a -(-2) 2(-1) 2 -2 Vertex: x = = = = -1 Axis of Symmetry x = -1 x y y = -(-1)2 - 2(-1) + 4 Vertex (-1, 5) = 5 x y Left Vertex -1 5 Right

Draw a curve through the points Example 5: Graph y = -x2 - 2x + 4 Second point: Choose x = 1 Axis of Symmetry y = -(1)2 – 2(1) + 4 = 1 x = -1 x y Vertex (-1, 3) x y Left Vertex -1 3 Right 1 -3 1 (-3, 1) (1, 1) Left of Vertex Right of Vertex Draw a curve through the points

Classwork y = -x2 + 2x + 2 −b 2a −( ) 2( ) −2 x = = = = 1 5) Second point: Choose x = 3 a = −1 b = 2 c = 2 y = -(3)2 + 2(3) + 2 = −1 −b 2a −( ) 2( ) 2 −2 Third point: mirror image of (3,-1) x = = = = 1 −1 y = −( )2 + 2( ) + 2 = 3 1 1 x y Left Vertex 1 3 Right −1 −1 3 −1

Classwork y = 2x2 + 8x + 6 −b 2a −( ) 2( ) −8 4 x = = = = −2 6) Second point: Choose x = 0 a = 2 b = 8 c = 6 y = 2(0)2 + 8(0) + 6 = 6 −b 2a −( ) 2( ) −8 4 8 x = = = = −2 Third point: mirror image of (0, 6) 2 y = 2( )2 + 8( ) + 6 = −2 -2 -2 x y Left Vertex -2 Right −4 6 6

-a -c Quadratic Equation Quadratic Term Linear Term Constant Term a c 7) How do the values of a, b, and c effect a parabola? Quadratic Equation Quadratic Term Linear Term Constant Term a c +a opens up y-intercept -a opens down +c shifts up -c skinny parabola shifts down wide parabola