Section 3.1: Quadratic Functions 1 MAT121: SECTION 3.1 QUADRATIC FUNCTIONS ANATOMY OF A QUADRATIC FUNCTION: GRAPH • Shape: Parabola – it may open upwards or down. • Vertex: Represented as (ℎ, ). It can be the maximum or minimum point depending on how the parabola opens. • Axis of Symmetry: Pass through the vertex. Always represented as the equation, = ℎ. Quadratic functions can be expressed in two ways: • Polynomial Form: = ! + + • Vertex Form: = − ℎ ! + VERTEX FORM = − ℎ ! + Note its similarity to the transformations from Section 2.6 • The sign of a determines the direction in which the parabola opens and whether the vertex is a maximum or minimum point. o a > 0 (positive) then the parabola opens upward and the vertex is a minimum. o a < 0 (negative) then the parabola opens downward and the vertex is a maximum. • In this form, the vertex is easily determined. (ℎ, ) are the coordinates of the vertex.
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MAT121: SECTION 3.1 QUADRATIC FUNCTIONS OF A QUADRATIC FUNCTION: GRAPH • Shape: Parabola – it may open upwards or down. • Vertex: Represented as (ℎ,!). It can be the maximum
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Section 3.1: Quadratic Functions 1
MAT121: SECTION 3.1
QUADRATIC FUNCTIONS
ANATOMY OF A QUADRATIC FUNCTION: GRAPH
• Shape: Parabola – it may open upwards or down.
• Vertex: Represented as (ℎ, 𝑘). It can be the maximum or minimum point depending on how the
parabola opens.
• Axis of Symmetry: Pass through the vertex. Always represented as the equation, 𝑥 = ℎ.
Quadratic functions can be expressed in two ways:
• Polynomial Form: 𝑓 𝑥 = 𝑎𝑥! + 𝑏𝑥 + 𝑐
• Vertex Form: 𝑓 𝑥 = 𝑎 𝑥 − ℎ ! + 𝑘
VERTEX FORM
𝑓 𝑥 = 𝑎 𝑥 − ℎ ! + 𝑘 Note its similarity to the transformations from Section 2.6
• The sign of a determines the direction in which the parabola opens and whether the vertex is a
maximum or minimum point.
o a > 0 (positive) then the parabola opens upward and the vertex is a minimum.
o a < 0 (negative) then the parabola opens downward and the vertex is a maximum.
• In this form, the vertex is easily determined. (ℎ, 𝑘) are the coordinates of the vertex.
Section 3.1: Quadratic Functions 2
Consider 𝑓 𝑥 = 𝑥 − 1 ! + 2
• Does the parabola open up or down? Why? Up, because a is positive.
• State the vertex. (1, 2)
• Is this a max or min? Minimum
• State the Axis of Symmetry. 𝑥 = 1
• State the Domain and Range. 𝐷𝑜𝑚𝑎𝑖𝑛: −∞,∞ 𝑅𝑎𝑛𝑔𝑒: [2,∞)
Consider 𝑓 𝑥 = 𝑥 + 4 ! + 1.
The function can be rewritten to fit the model. 𝑓 𝑥 = 𝑥 − −4 ! + 1
• State the vertex. (−4, 1)
• Is this a max or min? Minimum
• State the Axis of Symmetry. 𝑥 = −4
• Graph the function. What are the x- and y-intercepts?
x-intercepts: Consider the vertex. It’s above the x-axis and is the minimum, so no x-intercepts.
y-intercept: 𝑓 0 = 0+ 4 ! + 1 = 17 so the y-intercept is (0, 17)
ON YOUR OWN
Consider 𝑓 𝑥 = −2 𝑥 + 6 ! − 5.
• Does the parabola open up or down? Down
• State the vertex. Is it a max or min? Max
• State the y-intercept. (0,−77)
• State the Domain and Range. 𝐷: −∞,∞ 𝑅: (−∞,−5]
• Graph it.
Section 3.1: Quadratic Functions 3
POLYNOMIAL FORM
𝑓 𝑥 = 𝑎𝑥! + 𝑏𝑥 + 𝑐
• The sign of a determines the direction in which the parabola opens and whether the vertex is a
maximum or minimum point.
Consider 𝑓 𝑥 = 3𝑥! − 6𝑥 + 1.
• Does the parabola open up or down? Up, because the coefficient on x2 is positive.
• State the y-intercept. 𝑓 0 = 1, so the y-intercept is (0,1).
• Convert the function to vertex form by completing the square.
3𝑥! − 6𝑥 + 1
3(𝑥! − 2𝑥 )+ 1
3 𝑥! − 2𝑥 + 1 + 1− (3 ∙ 1)
3 𝑥 − 1 ! − 2
• State the vertex. (1,−2) It’s a minimum.
• Find the x-intercepts. Recall (𝑥, 0) represents the x-intercept.
Using the vertex form.
3 𝑥 − 1 ! − 2 = 0
3 𝑥 − 1 ! = 2
𝑥 − 1 ! =23
𝑥 − 1 = ±23
𝑥 = 1±23
Using the polynomial form and DQUAD.
3𝑥! − 6𝑥 + 1
𝑎 = 3
𝑏 = −6
𝑐 = 1
𝑥 = − − 6 ± 24
6
𝑥 = 6 ± 2 6
6 = 3± 63
Section 3.1: Quadratic Functions 4
FINDING AN EXPRESSION FOR VERTEX
Converting the polynomial model to the vertex model yields (page 323)
𝑓 𝑥 = 𝑎 𝑥 +𝑏2𝑎
!
+ 𝑐 −𝑏!
4𝑎
where ℎ!!!""#$%&'() !" !"#$"% = − !!!
and 𝑘!!!""#$%&'() !" !"#$"% = 𝑐 − !!
!!
Make it easier!
Given 𝑓 𝑥 = 𝑎𝑥! + 𝑏𝑥 + 𝑐, the x-coordinate of the vertex is found from – !!!
.
The y-coordinate of the vertex is found from 𝑓 – !!!