1 of 18 pre-cal chapter 2 section 1 sat/act warm - up
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Pre-Cal Chapter 2 Section 1
SAT/ACT Warm - up
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Pre-Cal Chapter 2 Section 1
Chapter 2 – Vocabulary Words
1. Polynomial function2. Linear function3. Quadratic function4. Continuous5. Intermediate Value Theorem6. Synthetic division7. Imaginary number8. Complex conjugates9. Rational function10. Slant Asymptote
See Page 172
Graphing Quadratic Functions
Chapter 2 Sec 1
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Essential Question
How do you sketch graphs and write
equations of parabolas?
Key Vocabulary:Polynomial
FunctionLinear equationQuadratic function
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Definitions
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Quadratic function
The graph of a quadratic function is a parabola.Every parabola is symmetrical about a line called the axis (of symmetry).
The intersection point of the parabola and the axis is called the vertex of the parabola.
x
y
axis
f (x) = ax2 + bx + cvertex
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Leading Coefficient
The leading coefficient of ax2 + bx + c is a.
When the leading coefficient is positive, the parabola opens upward and the vertex is a minimum.
When the leading coefficient is negative, the parabola opens downward and the vertex is a maximum.
x
y
f(x) = ax2 + bx + ca > 0 opens upward
vertex minimum
x
y
f(x) = ax2 + bx + c
a < 0 opens downward
vertex maximum
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5
y
x
5-5
Simple Quadratic Functions
The simplest quadratic functions are of the form f (x) = ax2 (a 0) These are most easily graphed by comparing them with the graph of y = x2.
222 2 and 2
1 , xxgxxfxy
2
2
1xxf
22xxg
Example: Compare the graphs of
2xy
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Example: f(x) = (x –3)2 + 2
Example: Graph f (x) = (x – 3)2 + 2 and find the vertex and axis.
f (x) = (x – 3)2 + 2 is the same shape as the graph ofg (x) = (x – 3)2 shifted upwards two units. g (x) = (x – 3)2 is the same shape as y = x2 shifted to the right three units.
f (x) = (x – 3)2 + 2
g (x) = (x – 3)2y = x 2
- 4x
y
4
4
vertex (3, 2)
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Completing the Square
1. Group together the x’s
2. Factor out the a.
3. Take coefficient of x.
a. Divide by 2
b. Then square result
4. Add the “squared result” inside the parenthesis and subtract the “Squared result times a” outside the parenthesis.
5. Factor the parenthesis, it will be (x ± (3a.))2.
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x
y
Quadratic Function in Standard Form
Example: Graph the parabola f (x) = 2x2 + 8x + 5 and find the axis and vertex.
f (x) = 2x2 + 8x + 5 original equation
f (x) = 2( x2 + 4x) + 5 group the x’s & factor out 2
f (x) = 2( x2 + 4x + 4) + 5 – 8 complete the square
f (x) = 2( x + 2)2 – 3 standard form
a > 0 parabola opens upward like y = 2x2.
h = –2, k = –3 axis x = –2, vertex (–2, –3). x = –2
f (x) = 2x2 + 8x + 5
(–2, –3)
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x
y
4
4
Vertex and x-Intercepts
Example: Graph and find the vertex and x-intercepts of f (x) = –x2 + 6x + 7.
f (x) = – x2 + 6x + 7 original equation
f (x) = – ( x2 – 6x) + 7 factor out –1
f (x) = – ( x2 – 6x + 9) + 7 + 9 complete the square
f (x) = – ( x – 3)2 + 16 standard form
a < 0 parabola opens downward.
h = 3, k = 16 axis x = 3, vertex (3, 16).
Find the x-intercepts by solving–x2 + 6x + 7 = 0.
(–x + 7 )( x + 1) = 0 factor
x = 7, x = –1 x-intercepts (7, 0), (–1, 0)
x = 3f(x) = –x2 + 6x + 7
(7, 0)(–1, 0)
(3, 16)
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y
x
f (x) = a(x – h)2 + k standard form
f (x) = a(x – 2)2 + (–1) vertex (2, –1) = (h, k)
y = f(x)
(0, 1)
(2, –1)
Since (0, 1) is a point on the parabola: f (0) = a(0 – 2)2 – 1
1 = 4a –1 and 2
1a
122
112
2
1 22 xxxfxxf
Example: Find an equation for the parabola with vertex (2, –1) passing through the point (0, 1).
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Example: Find the vertex of the graph of f (x) = x2 – 10x + 22.
f (x) = x2 – 10x + 22 original equation
a = 1, b = –10, c = 22
At the vertex,
So, the vertex is (5, -3).
a
bf
a
b
2,
2 is
The vertex of the graph of f (x) = ax2 + bx + c (a 0)
512
10
2
a
bx
Vertex of a Parabola
322510552
2
f
a
bf
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Minimum and Maximum Values of Quadratic Functions
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The path is a parabola opening downward. The maximum height occurs at the vertex.
At the vertex,
So, the vertex is (9, 15).
The maximum height of the ball is 15 feet.
629
1 2 xxy
2,9
162
9
1 2
baxxy
.9
91
2
2
2
a
bx
1592
f
a
bf
Example: A basketball is thrown from the free throw line from a height of six feet. What is the maximum height of the ball if the path of the ball is:
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Essential Question
How do you sketch graphs and write
equations of parabolas?
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Pre-Cal Chapter 2 Section 1
Daily Assignment
• Chapter 2 Section 1/ Ch 1 Review• Text Book (TB)
• Textbook Pgs 99– 102 #1 – 4, 7, 11, 17, 21,
27, 29, 31, 35, 55, 63 – 67 All
• Read Sections 2.2 – 2.3
• Show all work for credit.