QUADTRATIC RELATIONS

A relation which must contain a term with x2 It may or may not have a term with x and a constant term (a term without x) It can be written in 3 forms: 1.Standard form 2.Vertex form 3.Zeros form

y= ax 2 + bx + c Where a, b, and c are any number

y= a (x-h) 2 + k Where a, h, and k are any number

y= a(x-b)(x-c) Where a, b, and c are any number

A QUADRATIC RELATION ALWAYS HAS THE SHAPE OF A PARABOLA WHICH CAN OPEN UP OR DOWN (SEE NEXT SLIDE)

THE LINE WHICH SEEMS TO CUT IN HALF IS CALLED THE LINE OF SYMMETRY

THE VERTEX IS THE POINT WHICH IS THE HIGHEST OR LOWEST POINT ON THE GRAPH

OPENS UP- WHEN A > 0 OPENS DOWN- WHEN A < 0

CHARACTERISTICS OF A QUADRATIC RELATION WRITTEN IN VERTEX FORM y= a (x-h) 2 + k LINE OF SYMMETRY To find the line of symmetry, the expression in the brackets equals 0. Then solve for x. If there are no brackets, the line of symmetry is x=0. Note the value of h is always the opposite value (negative of) the number inside the bracket. (the sign of h is always the opposite of the operation in the brackets). If the x intercept is not 0, it will always be the same number in the bracket but have the OPPOSITE sign.

EXAMPLES: A) y= -2 (x+3) 2 – 4 LINE OF SYMMETRY IS x+3=0 or x=-3 B) y= 5 (x-1) LINE OF SYMMETRY IS x-1=0 or x=1 C) y= 0.5x LINE OF SYMMETRY IS x=0 D) y= 2(x-7) 2 LINE OF SYMMETRY IS x-7=0 or x=7

VERTEX To find the vertex, the x coordinate must be the same as the line of symmetry, and the y coordinate must be the last number (k) A) y=-2(x+3) 2 -4 LINE OF SYMMETRY IS x=-3, Vertex is (-3, -4) B) y=5(x-1) 2 +2 LINE OF SYMMETRY IS x=1, Vertex is (1, 2) C) y=0.5x 2 +5 LINE OF SYMMETRY IS x=0, Vertex is (0, 5) D) y=-2(x-7) 2 LINE OF SYMMETRY IS x=7, Vertex is (7, 0) (h, k) Line of symmetry Last number

GRAPHING QUADRATICS IN VERTEX FORM Step 1: determine the line of symmetry Step 2: Calculate the y coordinates of the vertex Step 3: draw x, y grid, label the x and y axis, and then plot the vertex Step 4: Now use a table of values to determine points near the vertex (2 numbers before and 2 numbers after the line of symmetry) Step 5: Plot the points from your table of values and join them with a curved line. Add arrows to each end and label the graph with the equation.

LAST CLASS

GRAPHING QUADRATICS IN STANDARD FORM Step 1: determine the line of symmetry and the vertex This is a little more difficult since you now have the term bx. A. To find the line of symmetry, you use the formula: B. To find the y value of the vertex, substitute this into the equation. y= ax 2 + bx + c

GRAPHING QUADRATICS IN STANDARD FORM B. To find the y value of the vertex, substitute this into the equation. Example: Determine the line of symmetry and the coordinates of the vertex. y= 4x 2 – 2x + 5 Line of symmetry : a=4, b=-2, c=5 y= ax 2 + bx + c

GRAPHING QUADRATICS IN STANDARD FORM B. To find the y value of the vertex, substitute this into the equation. y= 4x 2 – 2x + 5 At x=0.25, y= 4(0.25) 2 – 2(0.25) + 5 = =5 ∴, the coordinates of the vertex are (0.25, 5)

GRAPHING QUADRATICS IN STANDARD FORM COMPLETE QUESTION 4! Determine the line of symmetry and the coordinates of the vertex for a - f

STEPS FOR GRAPHING QUADRATIC EQUATIONS IN STANDARD FORM: Step 1: determine the line of symmetry Step 2: Calculate the y coordinates of the vertex Step 3: draw x, y grid, label the x and y axis, and then plot the vertex Step 4: Now use a table of values to determine points near the vertex (2 numbers before and 2 numbers after the line of symmetry) Step 5: Plot the points from your table of values and join them with a curved line. Add arrows to each end and label the graph with the equation.

GRAPHING QUADRATICS IN STANDARD FORM 1) Find the Line of Symmetry y= 3x 2 - 6x + 5 Identify the variables a=3, b=-6, c=5 Substitute into the formula: y= ax 2 + bx + c

GRAPHING QUADRATICS IN STANDARD FORM 2) Find the Coordinates of the Vertex At x=1, y= 3(1) 2 - 6(1) + 5 =3 – 6 +5 =2 ∴, the coordinates of the vertex are (1, 2) y= 3x 2 - 6x + 5

(1, 2) y= 3x 2 - 6x + 5 3) Graph the vertex

4. USE A TABLE OF VALUES TO DETERMINE POINTS NEAR VERTEX Xy Use a table of values with x=1,2,3 and x=0,-1 to find more points close to the vertex. y= 3x 2 - 6x + 5 =3(-1) 2 - 6(-1) + 5 =13 y= 3x 2 - 6x + 5 =3(0) 2 - 6(0) + 5 =4

STEPS FOR GRAPHING QUADRATIC EQUATIONS IN STANDARD FORM: Step 1: determine the line of symmetry Step 2: Calculate the y coordinates of the vertex Step 3: draw x, y grid, label the x and y axis, and then plot the vertex Step 4: Now use a table of values to determine points near the vertex (2 numbers before and 2 numbers after the line of symmetry) Step 5: Plot the points from your table of values and join them with a curved line. Add arrows to each end and label the graph with the equation.

GRAPHING QUADRATICS IN STANDARD FORM 1) Find the Line of Symmetry y= -2x 2 + 8x - 3 Identify the variables a=-2, b=8, c=-3 Substitute into the formula: y= ax 2 + bx + c

GRAPHING QUADRATICS IN STANDARD FORM 2) Find the Coordinates of the Vertex At x=1, -2(2) 2 + 8(2) - 3 = =5 ∴, the coordinates of the vertex are (2, 5) y=-2x 2 + 8x - 3

(2, 5) y=-2x 2 + 8x - 3 3) Graph the vertex

4. USE A TABLE OF VALUES TO DETERMINE POINTS NEAR VERTEX Xy Use a table of values with x=2,3,4 and x=1,0 to find more points close to the vertex. y=-2(2) 2 + 8(2) - 3 = =5

(0, -3) (1, 3) (2, 5) (3, 3) (4, -5) y=-2x 2 + 8x - 3 5) Plot & connect the points, arrows and label

GRAPHING QUADRATICS IN STANDARD FORM COMPLETE QUESTION 5! Graph a-h quadratic relations using: 1) Find the line of symmetry 2) Calculate the y coordinates of the vertex 3) Graph the vertex 4) Use a table of values to determine points near the vertex.

OPENS UP- WHEN A > 0 OPENS DOWN- WHEN A < 0

OPTIMAL VALUE  The height of the highest or lowest point  Always the last number  That is the maximum value if the graph opens down  That is the minimum value if the graph opens up. The OPTIMAL VALUE always corresponds to the y coordinate of the vertex. To find the value of the optimal value:  A) Find the line of symmetry  B) find the vertex, by substitution (This is the optimal value)

y = ax 2 + bx + c The parabola will open down when the a value is negative. Opens DOWN- When A < 0 The parabola will open up when the a value is positive. OPENS UP- When A > 0 OPTIMAL VALUE The standard form of a quadratic function is: a > 0 a < 0 (minimum). If the parabola opens up, the lowest point is called the vertex (minimum). (maximum If the parabola opens down, the vertex is the highest point (maximum).

GRAPHING QUADRATICS IN STANDARD FORM COMPLETE QUESTION 6 and 7! Find the maximum and minimum values