Quadratic Functions
What is a quadratic formula?
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Polynomials whose highest degree is 2
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+ 6 + 2, − 3 + 4,9 + are examples of quadratic functions
General Form: + + where ≠ •
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If > 0, If < 0, "#
Finding The Vertex (%, &)
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Highest or lowest point of the graph
To find ℎ:
% = − To find *:
& = +(%)
* Find * by plugging ℎ into your given function
NOTE: If asked for a maximum or minimum value, it will ALWAYS be the vertex because the vertex will
always be your highest or lowest point on the graph
Example 1 : Find the vertex of ,() = 2 + 8 + 7.
Solution: = 2, = 8, / = 7
ℎ=−
1
2
vertex is (−2, −1)
3
= − = −2
* = ,(ℎ) = ,(−2)
= 2(−2) + 8(−2) + 7
= 8 − 16 + 7
= −1
Standard Form: +() = ( − %) + &
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Standard form is used when you have vertex (ℎ, *) and ( is usually the coefficient of )
Example 2: Standard form of example 1 is as follows:
,() = ( − ℎ) + *
,() = 2( − (−2)) + (−1)
,() = 2( + 2) − 1
Graphing Polynomials
Leading Coefficient Test
Given the function ,() = 7 7 + ⋯ + + 9
Follow these steps to graph polynomials:
1. Use Leading Coefficient Test to determine the behavior of the ends of the graph.
2. Find the real zeros (in other words, find x-intercepts which means set 4 = 0).
3. Use factored form of function to determine multiplicity of each term (exponent of each term).
a. If multiplicity is odd, graph crosses the x-axis at ) ( is each of the real zeros).
b. If multiplicity is even, graph touches the x-axis at ) .
4. Determine turning points of graph (= highest degree of function turning points =5 − 6)
5. Graph real zeros on the graph, then pick a test point in between each graphed point to
determine the behavior throughout the graph
Congratulations, you have successfully graphed a polynomial!!! Way to go!