4.10 Write Quadratic Functions and Models

Best-fitting quadratic model

The model given by performing quadratic regression on a calculator

Example 1

Write a quadratic function in vertex form

Write a quadratic equation for the parabola shown.

y = a(x - h)² + k

Use the vertex form because the vertex is given.

y = a(x + 2_)² _- 3_ Substitute.

Use the other given point, (_0_ , _5_), to find a.

_5_ = a(_0 + 2_)² _- 3_ Substitute for x and y.

_2 = a _ Solve for a.

A quadratic function for the parabola is

_y = 2(x + 2)² - 3_ .

Example 2

Write a quadratic equation for the parabola shown.

y = a(x - p)(x - q)

Use the intercept form because the x-intercepts are given.

y = a(x + 3_)(x - 2_) Substitute.

Use the other given point, (_-2_ , _-4_), to find a.

_-4_ = a(_-2 + 3_)(-2 - 2_) Substitute for x and y.

_1_ = a Solve for a.

A quadratic function for the parabola is

_y = (x + 3)(x- 2) .

Example 3

Write a quadratic function in standard form

Write a quadratic function in standard form for the parabola that passes through the points (-2, -6), (0, 6) and (2, 2).

Substitute the coordinates of each point into y = ax² + bx + c to obtain a system of three linear equations.

_-6_ = a(_-2_)^2 + b(_-2_) + c	Substitute for x and y.
_-6_ = _4a - 2b + c	Equation 1
_6_ = a(_0_)^2 + b(_0_) + c	Substitute for x and y.
_6_ = _c_	Equation 2
_2_ = a(_2_)^2 + b(_2_) + c	Substitute for x and y.
_2_ = _4a + 2b + c_	Equation 3
Rewrite the system as a system of two equations.
_4a - 2b + 6_ = _-6_	Substitute for c.
Substitute 6 for c in equation 1.
_4a - 2b = _-12_	Revised Equation 1
_4a + 2b + 6_ = _2_	Substitute for c.
Substitute 6 for c in equation 3.
_4a + 2b = _-4_	Revised Equation 3
Solve the system consisting of revised equations 1 and 3.
_4a - 2b = -12_	Revised Equation 1
_4a + 2b = -4_	Revised Equation 3
_8a _= -16_	Add Equations.
a = _-2_	Solve for a.
So _4(- 2) + 2b = _-4_, which means b = _2_ .
A quadratic function for the parabola is
_y = -2x^2 + 2x + 6_.

Example 4

Solve a multi-step problem

Baseball The table shows the height of a baseball hit, with x representing the time
(In seconds) and y representing the baseball’s height (In feet). Use a graphing calculator to find the best-fitting model for the data.

Time, x	0	2	4	6	8
Height, y	3	28	40	37	26

Solution

Enter the data into two lists of a graphing calculator.	Make a scatter plot of the data.
Use the quadratic regression model feature to find the best-fitting quadratic model for the data.	Check how well the model fits the data by graphing the model and the data in the same viewing window.
QuadReg y = ax^2 + bx + c a = _-1.553571429_ b = _15.17857143_ c = _3.371428571_
The best fitting quadratic model is _y » -1.55x^2 + 15.2x + 3_.