Chapter 4 Section 4-10 Writing Equations: 1. Vertex Point plus another Point 2. Quadratic Regression.

Chapter 4

Section 4-10

Writing Equations:

1. Vertex Point plus another Point

2. Quadratic Regression

Objectives

• I can write the equation of a quadratic function given the vertex and another point

• I can find the equation to any quadratic function given at least 3 points

Vertex Format

• Recall that vertex format for any quadratic equation is given by:

• y = a (x – h)2 + k

• Where (h, k) is the vertex point

Finding the Equation

• If we know the vertex of a parabola, then we know h and k

• Then if we know any other point, we know an x and y value, so we can solve for “a” in the equation

Writing an Equation in Vertex Format

2( )y a x h k

ex: ( , )Vert h kint : ( , )Po x y

Example 1

• Find the equation of the parabola with vertex (3, -4) and goes through point (0, 1)

• y = a(x – h)2 + k

• There are 5 variables here: a, h, k, x, y• We know 4 of the 5, so we can solve for “a”

Vertex (3, -4) and Point (0, 1)

7

2( )y a x h k 21 (0 3) 4a

1 9 4a

5

9a

25( 3) 4

9y x 5 9a

Vertex (3, 1) and Point (6, 3)

8

2( )y a x h k 23 (6 3) 1a

3 9 1a

2

9a

22( 3) 1

9y x

9

Example 3• Write the equation in Standard Vertex Format for

the parabola that passes thru point (2, 1) and has a vertex point (-2, -3)

• y = a(x – h)2 + k• 1 = a(2 - -2)2 + -3• 1 = a(4)2 – 3• 1 = 16a – 3• 4 = 16a• a = 1/4

3)2(4

1 2 xy

Example 4• Write the equation in Standard Vertex Format for

the parabola that passes thru point (-3, -5) and has a vertex point (4, 1)

• y = a(x – h)2 + k• -5 = a(-3 - 4)2 + 1• -5 = a(-7)2 + 1• -5 = 49a + 1• -6 = 49a• a = -6/49

26( 4) 1

49y x

Quadratic Regression

• If we know 3 points on a parabola, then we can use quadratic regression and find the equation using the calculator

• Given we know the following 3 ordered pairs that are on the parabola

• (1, 2), (4, 3), and (7, 6)• Write the equation : y = ax2 + bx + c

STAT Button

• 3rd row

• 3rd button

Entering Data

STAT Button

• 3rd row

• 3rd button

Calculating the Equation

Possible Errors

• Enter wrong number into L1 or L2

• Forget a negative sign

• Forget to clear L1 or L2

• Use a list other than L1 and L2

Example 1

• Find the equation of the parabola that passes through the following points

• (-1, -1), (1, 11), (3, 7)• STAT• EDIT• Enter the L1 data {-1, 1, 3}• Enter L2 data {-1, 11, 7}• Now STAT to Calc #5 Quad Reg then ENTER• y = -2x2 + 6x + 7

Homework

• WS 6-5