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M3U4D11 Warm Up
A rocket is launched into the air so that
its height, in feet, after t seconds is
modeled by the equation:
h(t) = -16t2+75t+50
When and what is the maximum height
of the rocket and when will it hit the
ground?
Max height 137.891 feet at 2.344 sec.
and it will hit the ground at 5.279 sec.
Homework Check:
ON DOCUMENT CAMERA
REMEMBER:
Standard Form
y = ax2 + bx + c
and
Vertex Form
y = a(x-h)2 + k
QUIZ !!!
M3U4D10b Graphing
Polynomial Functions
OBJ: Use the process of
completing the square in a
quadratic function to show
zeros, extreme values, and
symmetry of the graphs.
M3U4D11 Systems of NonLinear Equations
OBJ: Solve polynomial equations
and systems of polynomial
equations approximately by using
technology to graph the functions
they define. (A-REI.11)
Systems of Nonlinear Equations and Their
Solutions
A system of two nonlinear equations in two variables contains at least one
equation that cannot be expressed in the form Ax + By = C. Here are two
examples:
x2 = 2y + 10
3x – y = 9
y = x2 + 3
x2 + y2 = 9
A solution to a nonlinear system in two variables is an ordered pair of real
numbers that satisfies all equations in the system. The solution set to the
system is the set of all such ordered pairs.
Example 1 – Substitution: Two-Solution Case
• Solve the system of equations.
•
3x2 + 4x – y = 7 Equation 1
•
2x – y = –1 Equation 2
• Solution:
• Begin by solving for y in Equation 2 to obtain y = 2x +
1.
• Next, substitute this expression for y into Equation 1
and solve for x.
•
•
3x2 + 4x – (2x + 1) = 7 Substitute 2x + 1 for y in Equation 1.
3x2 + 2x – 1 = 7 Simplify.
Example 1 – Solution
•
3x2 + 2x – 8 = 0
•
(3x – 4)(x + 2) = 0
cont’d
Write in general form.
Factor.
Solve for x.
• Back-substituting these values of x to solve for the
• corresponding values of y produces the solutions
• and
Graphical Approach to Finding
Solutions
• A system of two equations in two unknowns can
have exactly one solution, more than one solution,
or no solution.
• By using a graphical method, you can gain insight
about the number of solutions and the location(s) of
the solution(s) of a system of equations by graphing
each of the equations in the same coordinate
plane.
• The solutions of the system correspond to the points
of intersection of the graphs.
Graphical Approach to Finding
Solutions
• For instance, the two equations in Figure 6.1 graph as two lines
with a single point of intersection; the two equations in Figure
6.2 graph as a parabola and a line with two points of
intersection; and the two equations in Figure 6.3 graph as a
line and a parabola that have no points of intersection.
One intersection point
Figure 6.1
Two intersection points
Figure 6.2
No intersection points
Figure 6.3
M3U4D11 Systems of Nonlinear Equations Notes p. 35
COMPLETE TOGETHER
Classwork 2
M3U4D11 Systems of Nonlinear Equations Notes #1-10
P. 36-37
Homework
Finish CW