systems of equations - review

8/7/2019 Systems of Equations - Review

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Systems of EquationsMiss Connell



R eview QuadraticsMiss Connell ² April 2011



DefinitionO



Forms of R ulesO



Properties1. The graph is called a parabola.

2. If a > 0, the parabola opens upward;

If a < 0, the parabola opens downward.

3. As |a| increases, the parabola becomes narrower;

As |a| decreases, the parabola becomes wider

4. The lowest point of a parabola (when a > 0)

or the highest point (when a < 0) is called the vertex.

5. The domain is (-� , �)The range is [k, �) when a > 0 or (�, k] when a < 0.

6. The graph is symmetric with respect to the axis of

symmetry. The equation of the axis of symmetry is x = h.

(h, k) is the vertex



V ertex and InterceptsO



Examplesa) f(x) = -2(x ² 3)2 + 2

b) f(x) = x2 - 8x + 10

c) f(x) = 2x2 + 12x + 17

O For each of the above, find the vertex, axis,

domain, range, and intercepts



Example Solutionsa) f(x) = -2(x ² 3)2 + 2

O Vertex (3, 2)

O

Axis of Symmetry x = 3O Domain (-�, �)

O a < 0, Range is (-�, 2]

O Y-intercept f(0) = -2(0 ² 3)2 + 2 = -16; (0, -16)



Example Solutionsa) f(x) = -2(x ² 3)2 + 2

O X-intercept 0 = -2(x ² 3)2 + 2

Square the binomial: -2(x2 ² 6x + 9) + 2 = 0

Simplify -2x2 + 12x - 18 + 2 = 0-2x2 + 12x - 16 = 0

Divide both sides by -2 x2 - 6x + 8 = 0Solve by factoring (x-4)(x-2) = 0

x -4 = 0 OR x ² 2 = 0

x = 4 x = 2The x-intercepts are (4,0) and (2,0)

O a < 0, the parabola opens downward,

O Since |a|=|-2|= 2, the parabola is narrower than f(x) = x2



Example Solutionsb) f(x) = x2 - 8x + 10

O Vertex;Group terms containing x f(x) = (x2 - 8x) + 10

Complete square by +16 f(x) = (x2 - 8x + 16 - 16) + 10half of -8 is -4; (-4)2 = 16also -16 to keep balanceRegroup f(x) = (x2 - 8x + 16) - 16 + 10Factor the perfect square f(x) = (x - 4)2 - 6

(4, -6)

O Axis of Symmetry x = 4

O Domain (-�, �)

O a > 0, Range is [-6, �)

O Y-intercept x = 0, f(0) = 02 ² 8(0) + 10 = 10 (0, 10)



Example SolutionsO



Maximum and MinimumO



ExampleO



Finding the R uleO



Lets Practice



Solving by FactoringO




1 242 123 84 6



Try a FewO O



O Turn this into logic

O Ask yourself O Two numbers

O Product equals ac

O Sum equals b




O Ask yourself

O Two numbers

O

Product equals (3)(8) = 24O Sum equals (-10)

1 242 123 84 6




O Ask yourself

O Two numbers

O

Product equals (6)(-10) = - 60O Sum equals (7)

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60



Try a FewO O



Now onto



Linear Equations

Express using

O TableO Rule

O Graph



ExampleO Consider that the daily cost to rent a car is

$60.00 and that the base fee is $25.00.

Represent this situation using a table, ruleand graph.



SolutionO

# days (x) Cost (y)

0 25

1 85

2 145

3 205



Straight Lines



Calculate unknown variableO



Straight Lines as Linear EquationsO



Example

O O

Find the rate of change and the initial

value for each of the following



If A(x1

,y 1

) and B(x2

,y 2

) are twopoints on a line then:

O



Find the R uleA straight line passing through points A(1,4) and B(-2,7)



Find the R uleA straight line passing through points A(-2,-3) & B(-8,4)



W ord ProblemsO A video store offers movie rentals. It costs

$8.50 to rent two movies and $17.50 to rent

5 movies.a) Find the rule that represents this situation

b) Find the cost of renting 7 movies

c) How many rentals would cost $11.50



Graphing Linear EquationsO Two methods:

1. Using a table of values2. Using the rate of change

and initial value



ExampleO

-3

-2

-1

0

1

2

3



ExampleO

systems of equations - review

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