Modeling With Quadratic Functions Section 2.4 beginning on page 76
Dec 18, 2015
Modeling With Quadratic Functions
Section 2.4 beginning on page 76
The Big IdeasIn this section we will learn the different ways we can use quadratic functions to represent real world situations.
Core VocabularyAverage rate of change
System of three linear equations
Different Forms of Quadratic Equations
Given the Vertex and a PointExample 1: The graph shows the parabolic path of a performer who is shot out of a cannon, where y is the height (in feet) and x is the horizontal distance traveled (in feet). Write an equation of the parabola. The performer lands in a net 90 feet from the cannon. What is the height of the net?
Given: Vertex: Point:(50,35) (0,15)
Substitute the values from the vertex and the point into vertex form.
π¦=π(π₯βh)2+π35
15=2500π+35β20=2500πβ0.008=π
π¦=π(π₯βh)2+ππ¦=β0.008(π₯β50)2+35
Now find the height when x = 90 (where the performer lands)
π¦=β0.008(90β50)2+35π¦=22.2
The height of the net is 22 feet.
Given the Vertex and a Point2) Write an equation of the parabola that passes through the point (-1,2) and has a vertex of (4,-9)
Given: Vertex: Point:(4 ,β9) (β1,2)
Substitute the values from the vertex and the point into vertex form.
π¦=π(π₯βh)2+π
2=π(β1β4)2β9
2=25πβ911=25 π
0.44=π
π¦=π(π₯βh)2+ππ¦=0.44 (π₯β4 )2β9
2=π(β5)2β9
Given a Point and the x-interceptsExample 2: A meteorologist creates a parabola to predict the temperature tomorrow, where x is the number of hours after midnight and y is the temperature (in degrees Celsius).
a) Write a function f that models the temperature over time. What is the coldest temperature?
π¦=π(π₯βπ)(π₯βπ)9.6=π(0β4)(0β24)9.6=96π.1=π
π¦=π(π₯βπ)(π₯βπ)π¦=.1(π₯β4 )(π₯β24 )
The lowest temp is the minimum.
b) What is the average rate of change in temperature over the interval in which the temperature is decreasing? Increasing? Compare the average rates of change.
π₯=4+242 ΒΏ14
π¦=.1(14β4)(14β24)π¦=β10 -10c is the lowest temp.
The interval before the minimum is decreasing, and the interval after the minimum is increasing. We can find the average slope for each segment.(We will do this part on the board)
Given a Point and the x-intercepts
π¦=π(π₯βπ)(π₯βπ)5=π(2ββ2)(2β4 )5=β8π
β58=π
π¦=π(π₯βπ)(π₯βπ)
π¦=β58(π₯+2)(π₯β4)
4) Write an equation of the parabola that passes through the point (2,5) and has x-intercepts -2 and 4.
Quadratic Regression Continued
Practice Quadratic Regression