4.10 Write Quadratic Functions and Models

Goal Write quadratic functions and models.

Your Notes

VOCABULARY

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

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 function in intercept form

Write a quadratic equation for the parabola shown.

y  a(x  p)(x  q)

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_ = aSolve for a.

A quadratic function for the parabola is

_y =(x 3)(x2) .

Your Notes

CheckpointWrite a quadratic function whose graph has the given characteristics.

1.x-intercepts: 2, 1 point on graph: (1, 4)

y 2(x 2)(x1)

2.vertex: (2,1) point on graph: (0, 4)

y(x 2)2 1

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.
_4a 2b_12_ / Revised Equation 1
_4a 2b  6__2_ / Substitute for c.
_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2x2 2x 6_.

Your Notes

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ax2bxc
a_1.553571429_
b_15.17857143_
c_3.371428571_ /
The best fitting quadratic model is
_y1.55x2 15.2x 3_.

Your Notes

CheckpointComplete the exercises below.

3.Write a quadratic function in standard form for the parabola that passes through
(1, 5), (2, 1) and (3, 1).

yx2 3x 1

4.Use a graphing calculator to find the best-fitting model for the data in the table.

Time, x / 0 / 2 / 4 / 6 / 8
Height, y / 4 / 23 / 30 / 25 / 7

y1.54x2 12.7x 4.

Homework

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