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Systems of EquationsMiss Connell
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R eview QuadraticsMiss Connell ² April 2011
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DefinitionO
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Forms of R ulesO
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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
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V ertex and InterceptsO
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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
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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)
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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
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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)
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Example SolutionsO
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Example SolutionsO
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Example SolutionsO
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Maximum and MinimumO
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ExampleO
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ExampleO
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Finding the R uleO
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Finding the R uleO
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Lets Practice
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Solving by FactoringO
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Solving by FactoringO
1 242 123 84 6
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Solving by FactoringO
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Solving by FactoringO
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Try a FewO O
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Try a FewO O
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O Turn this into logic
O Ask yourself O Two numbers
O Product equals ac
O Sum equals b
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O Turn this into logic
O Ask yourself
O Two numbers
O
Product equals (3)(8) = 24O Sum equals (-10)
1 242 123 84 6
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O Turn this into logic
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
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Try a FewO O
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Now onto
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Linear Equations
Express using
O TableO Rule
O Graph
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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.
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SolutionO
# days (x) Cost (y)
0 25
1 85
2 145
3 205
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Straight Lines
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Calculate unknown variableO
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Straight Lines as Linear EquationsO
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Example
O O
Find the rate of change and the initial
value for each of the following
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If A(x1
,y 1
) and B(x2
,y 2
) are twopoints on a line then:
O
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Find the R uleA straight line passing through points A(1,4) and B(-2,7)
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Find the R uleA straight line passing through points A(-2,-3) & B(-8,4)
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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
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Graphing Linear EquationsO Two methods:
1. Using a table of values2. Using the rate of change
and initial value
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ExampleO
-3
-2
-1
0
1
2
3
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ExampleO
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ExampleO