Name __________________________________
Date ___________________
LESSON 1.1- 1.2 Study Guide
A quadratic function is a function that can be written in the standard form y = ax2 + bx + c where a  0. The
graph of a quadratic equation is a parabola. The lowest or highest point on a parabola is the vertex. The axis
of symmetry divides the parabola into mirror images and passes through the vertex. For y = ax2 + bx + c, the
vertex’s y-coordinate is the minimum value of the function if a > 0 and the maximum value if a < 0.
EXAMPLE 1
Graph a function of the form y = ax2 + c
Graph y = x2 + 2.
STEP 1 Make a table of values for y = x2 + 2.
x
2
1
0
1
2
y
2
1
2
1
2
STEP 2 Plot the points from the table.
STEP 3 Draw a smooth curve through the points.
EXAMPLE 2
EXAMPLE 2
Graph a function of the form y = ax2 + bx+ c
Graph y = x2  6x+ 8.
STEP 1 Identify the coefficients of the function:
a = 1, b =  6, c = 8.
STEP 2 Find the vertex. To find the x-coordinate,
use x =
=
. The y-coordinate is
2
(3)  6(3) + 8 = 1. Plot the vertex (3, 1).
STEP 3 Draw the axis of symmetry x = 3.
STEP 4 Identify the y-intercept (0, 8) and reflect
the point in the axis of symmetry to plot (6, 8).
STEP 5 Evaluate the function for x = 1. Plot the
point (1, 3) and its reflection (5, 3).
STEP 6 Draw a parabola through the plotted
points.
Exercises for Example1 & 2
Graph the function. Label the vertex and axis of symmetry.
1.
2.
3.
y = 6x2
y =  x2 + 1
y = 4x2  1
4.
5.
6.
y = x2 + 4x + 4
y = 2x2  8x + 6
y = x2  4x  4
EXAMPLE 3
Find the minimum or maximum value
Tell whether the function y = 2x2 + 8x  7 has a minimum value or a maximum value. Find the
minimum or maximum value.
-b
-8

Because a < 0, the function has a maximum value. The coordinates of the vertex are x =
=2
and y =  2(2)2 + 8(2) 7 = 1. The maximum value is y = 1.
2a
( 4 )
EXAMPLE 4
Find the maximum value of a quadratic function
Revenue A street vendor sells about 20 shirts each day when she charges $8 per shirt. If she decreases the
price by $1, she sells about 10 more shirts each day. How can she maximize daily revenue?
STEP 1 Define the variables. Let x represent the price reduction and R(x) represent the daily revenue.
STEP 2 Write a verbal model. Then write and simplify a quadratic function.
Revenue
(dollars)
=
Price
(dollars/shirt)

Number of
shirts
R(x)
=
(8  x)
 (20 + l0x)
R(x)
= 160  20x + 80x 10x2
R(x) = 10x2 + 60x+ 160
STEP 3 Find the coordinates (x, R(x)) of the vertex.
x=
b
60
=3

2a
2(10)
Find x-coordinate.
R(3) = 10(3)2 + 60(3) + 160 = 250
Evaluate R(3).
The vertex is (3, 250). The vendor should reduce the price by $3 to maximize daily revenue.
Exercises for Examples 3 and 4
Tell whether the function has a minimum value or a maximum value. Then find the minimum or
maximum value.
y = 2x2 + 8x  3
___________________ __________
2
y = x + 6x – 8
___________________ __________
2
y = x 4x  12
___________________ __________
Maximize Revenue A sports store sells about 50 mountain bikes per month at a price of $220 each.
For each $20 decrease in price, about 10 more bikes per month are sold.
10. Write a quadratic function in standard form that models the revenue from bike sales.__________
11. What price produces the maximum revenue?___________________
7.
8.
9.
LESSON 1.2
The vertex form of a quadratic equation is given by y = a(x  h)2 + k.
The intercept form of a quadratic equation is given by y = a(x  p)(x  q).
EXAMPLE 5
Graph a quadratic function
a. Graph y = 2(x  1)2 + 3, a
quadratic function in vertex form.
b. Graph y = 3(x  4)(x + 2),
a quadratic function in intercept form.
STEP 1: Identify a = 2, h = 1, and
k = 3. Because a  0, the
parabola opens up.
STEP 1: Identify the x-intercepts.
Because p = 4 and q = 2, (4, 0)
and (2, 0) are the x-intercepts.
STEP 2: Plot the vertex (1, 3) and
draw x = 1, the axis of
symmetry.
STEP 2: Find the coordinates of the vertex.
STEP 3: Evaluate y for x = 0 and x = 2.
Plot the points (0, 5) and (2, 5).
y = 3(1  4)(1 + 2) = 27
The Vertex is (1, 27)
STEP 4: Draw a parabola through the
Points
STEP 3: Draw a parabola through the vertex and the
points where the x-intercepts occur
x 
pq
2

4  (2)

2
2
1
2
Exercise for Example 5
Graph the function. Label the vertex, axis of symmetry, and the x-intercepts.
1. y = 2(x + 2)2
2. y = 2(x  2)(x  3)
3. y = (x + 4)(x + 2)
EXAMPLE 6: Use a quadratic model in vertex form
Bridges A bridge is designed with cables that connect two towers that rise above a roadway. Each cable is
modeled by
( x  1600 ) 2
y 
 30
6800
Find the distance between the towers.
Solution
The vertex of the parabola is (1600, 30). The cable's lowest point is 1600 feet from either tower. The distance
between the towers is d = 2(1600) = 3200 feet.
Exercise for Example 6
7. Rework Example 2 where
y 
( x  1500 ) 2
 28
__________________________
6500
Golf The flight of a particular golf shot can be modeled by the function
y =  0.001x(x  260) where x is the horizontal distance (in yards) from the impact point and y is the
height in yards. The graph is shown.
8. How many yards away from the impact point does the golf ball land?
____________________________
9. What is the maximum height in yards of the golf shot?
_______________________________
EXAMPLE 7: Change quadratic functions to standard form
Solution
Write the quadratic function in standard form.
a.
y = 3(x  l)(x + 2)
a. y = 3(x  l)(x + 2)
b.
Write original function
in intercept form.
2
= 3(x + 2x  x  2) Multiply using FOIL.
= 3(x2 + x  2)
Combine like terms.
= 3x2 + 3x  6
Distributive property
y = (x + 1)2 + 2
b.
y = (x + 1)2 + 2
Write original function
in vertex form.
= (x + l)(x + 1) + 2 Rewrite (x + 1)2.
= (x2 + x + x + l) + 2 Multiply using FOIL.
= x2 + 2x + 3
Combine like terms.
Exercise for Example 7: Write the quadratic function in standard form.
10.y = (x + 1)2
13.y = (x + l)(x  1)
__________________________
11.y = (x  1)2
__________________________
14.y = 2(x  2)(x  3)
__________________________
12.y = 2(x + 2)2
__________________________
15.y= (x + 3)(x + 2)
__________________________
___________________