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

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)

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

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

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

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