3.1 quadratic functions and models. quadratic functions a quadratic function is of the form f(x) =...
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3.1
Quadratic Functions and Models
Quadratic Functions
A quadratic function is of the form
f(x) = ax2 + bx + c,
where a, b, and c are real numbers, a ≠ 0.
The graph of a quadratic function is a parabola. The domain of a quadratic function is all real
numbers. These functions have a linear rate of change.
Vertex The maximum or minimum point of a parabola
Axis of symmetry The vertical line passing through the vertex
Leading coefficient In a quadratic function is this “a” (the coefficient of x2). When positive the graph opens up. When negative the graph opens down. Larger values of |a| result in a narrower parabola, smaller
values of |a| result in a wider parabola.
Vertex Form of a Quadratic Function Vertex Form
The parabolic graph of f(x) = a(x – h)2 + k has vertex (h,k). Graph opens up when a > 0, down when a < 0.
Examples Page 184 Identify f as being linear, quadratic,
or neither. If it is quadratic, identify the leading coefficient and evaluate f(-2). #2 f(x) = 1 – 2x + 3x2 #4 f(x) = (x2 + 1)2
#6 f(x) = 1/5 x2
Page 185 Identify the vertex and the leading coefficient. Then write the equation as f(x) = ax2 + bx + c #18 f(x) = 5(x + 2)2 – 5 #20 f(x) = ½(x + 3)2 – 5
Finding the vertex
Vertex Formula The vertex for the graph of f(x) = ax2 + bx + c with
a ≠ 0 is the point
,2 2
b bf
a a
Examples
Page 185 Use the vertex formula to determine the vertex of the graph of f. #26 f(x) = 2x2 – 2x + 1
#30 f(x) = -3x2 + x – 2
Completing the Square
y = x2 + 6x – 8
Examples
Page 185 Write the given equation in the form f(x) = (x – h)2 + k. #40 f(x) = x2 + 10x + 7
#50 f(x) = 6 + 5x – 10x2
Quadratic Regression on the Calculator Enter data into List 1 and List 2
Choose Stat Calc 5: quadreg enter enter
To have the data go directly into y = Before you press the second enter Choose vars y-vars function y1
Examples
Page 187 #98 a) Make a scatterplot of the data.
b) Find the values for a, h and k. Graph f(x) together with the data in the same viewing rectangle.
c) Approximate the undetermined value(s) in the table.
U.S. population in millions
Year 1800 1820 1840 1860 1870 1880
Population 5 10 17 31 ? 50
Year 1900 1920 1940 1960 1980 2000
Population 76 106 132 178 226 ?
Problem Solving
Page 186 #82 Match the physical situation with the graph of the
quadratic function that models it best.
Example
Page 187 #102 The cables that support a suspension bridge,
such as the Golden Gate Bridge, can be modeled by parabolas.
Suppose that a 300-foot long suspension bridge has towers at its ends that are 120 feet tall, as illustrated in the accompanying figure.
If the cable comes within 20 feet of the road in the center of the bridge, find the quadratic function that models the height of the cable above the road a distance of x feet from the center of the bridge.
120 ft
300 ft20 ft
3.2
Quadratic Equations and Problem Solving
Examples
Page 201 #2
#10
2 9 10 8x x
28 63 46x x
Quadratic formula
The solutions to the quadratic equation
ax2 + bx + c = 0, where a ≠ 0, are given by
x = b b ac
a
2 4
2
#16
#18
23( 5) 6 0x
23 1 10
4 2 2x x
The Discriminant
The discriminant is used to determine the number of real solutions to ax2 + bx + c =0. If b2 – 4ac > 0, there are two real solutions. If b2 – 4ac = 0, there is one real solution. If b2 – 4ac < 0, there are no real solutions.
Examples Page 202
a. Write the equation in standard form
b. Calculate the discriminant and determine the number of real solutions
c. Solve the equation.
#46
#58
#60
28 2 14x
24 6x x
(5 3) 1x x
Solve graphically
Page 202 #4222 4 1.595x x
Problem Solving
Page 203 #100 From 1984 to 1994 the cumulative number of
AIDS cases can be modeled by the equation
Where x represents years after 1984. Estimate the year when 200,000 AIDS cases had been diagnosed.
2( ) 3034 14,018 6400,C x x x
Page 204 #108 A rectangular pen for a pet is under construction
using 100 feet of fence.a. Determine the dimension that result in an area of 576
square feet.
b. Find the dimensions that give the maximum area.
3.3
Quadratic Inequalities
Solving Quadratic Inequalities Write in Standard Form Solve Use the boundary numbers to test points Use the table or graph to write your solution
Examples
Page 213 Solve each equation and inequality. Write the solution set for each inequality in interval notation. #12
a.
b.
c. #14
a.
b.
c.
2 8 12 0x x 2 8 12 0x x 2 8 12 0x x
2 17 0n 2 17 0n
2 17 0n
#16a.
b.
c.
#18a.
b.
c.
#22a.
b.
c.
27 4 0x x 27 4 0x x 27 4 0x x
2 2 1 0x x 2 2 1 0x x 2 2 1 0x x
22 4 3 0x x 22 4 3 0x x 22 4 3 0x x
3.4
Transformations of Graphs
Shifting and Stretching
Graph y1 = x2
y2 = x2 + 3 y3 = x2 – 3
What pattern do you see?
Vertical Shifts
Vertical Shifts g(x) = f(x) + a, shift graph up a units g(x) = f(x) – a, shift graph down a units
Graph y1 = x2
y2 = (x + 3)2
y3 = (x – 3)2
What pattern do you see?
Horizontal Shifts
Horizontal Shifts g(x) = f(x + a), shift graph left a units g(x) = f(x – a), shift graph right a units
Graph y1 = x2
y2 = 3x2
y3 = 6x2
Stretching
Vertical and Horizontal stretches: For a >0, the graph g(x) = af(x) stretches the
graph vertically by a factor of a. For a >1, the graph g(x) = f(ax) compresses the
graph horizontally by a factor of a. h(x) = f(x/a) compresses the graph horizontally by
a factor of a.
Graph y1 = x2
y2 = -x2
Negative Coefficients
When you multiply by a negative it reflects (flips) the graph over the x-axis.
Predict what will happen y = -x2 + 3
y = -2(x + 5)2 - 3
f(x) = x2, af(x + b) + c
Examples
Page 229 Use the accompanying graph of y = f(x) to sketch a graph of each equation. #12
a. y = f(x + 1)
b. y = -f(x)
c. Y = 2f(x) #14
a. y = f(x – 1) - 2
b. y = -f(x) + 1
c. y = f(1/2x)
Other Parent Graphs
y = x
y = |x|
y = x3
y = √x
Examples
Page 230 Use transformations for graphs to sketch a graph of f. #50
#52
#68
( ) 1f x x
( ) | 4 |f x x
3( ) ( ) 1f x x