Find the x-intercept and y-intercept
1. 3x – 5y = 15
2. y = 2x + 7
ANSWER (5, 0); (0, –3)
ANSWER ( , 0) ; (0, 7)72
–
Parabola Parent Function: 2xy
Graphing Quadratic Functions
y = ax2 + bx + c
Quadratic Functions
The graph of a quadratic function is a parabola.
A parabola can open up or down.
If the parabola opens up, the lowest point is called the vertex.
If the parabola opens down, the vertex is the highest point.
NOTE: if the parabola opened left or right it would not be a function!
y
x
Vertex
Vertex
minimum
maximum
y = ax2 + bx + c
The parabola will open down when the a value is negative.
The parabola will open up when the a value is positive.
Standard Form
y
x
The standard form of a quadratic function is
a > 0
a < 0
y
x
Line of Symmetry
Line of Symmetry
Parabolas have a symmetric property to them.
If we drew a line down the middle of the parabola, we could fold the parabola in half.
We call this line the line of symmetry.
The line of symmetry ALWAYS passes through the vertex.
Or, if we graphed one side of the parabola, we could “fold” (or REFLECT) it over, the line of symmetry to graph the other side.
Line of Symmetry
Find the line of symmetry of y = 3x2 – 18x + 7
Finding the Line of Symmetry
When a quadratic function is in standard form
The equation of the line of symmetry is
y = ax2 + bx + c,
2ba
x
For example…
Using the formula…
This is best read as …
the opposite of b divided by the quantity of 2 times a.
18
2 3x 18
6 3
Thus, the line of symmetry is x = 3.
Finding the Vertex
We know the line of symmetry always goes through the vertex.
Thus, the line of symmetry gives us the x coordinate of the vertex.
To find the y coordinate of the vertex, we need to plug the x value into the original equation.
STEP 1: Find the line of symmetry
STEP 2: Plug the x value into the original equation to find the y value.
y = –2x2 + 8x –3
8 8 22 2( 2) 4
ba
x
y = –2(2)2 + 8(2) –3
y = –2(4)+ 8(2) –3
y = –8+ 16 –3
y = 5
Therefore, the vertex is (2 , 5)
A Quadratic Function in Standard Form
The standard form of a quadratic function is given by
y = ax2 + bx + c
There are 3 steps to graphing a parabola in standard form.
STEP 1: Find the line of symmetry
STEP 2: Find the vertex
STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.
Plug in the line of symmetry (x value) to
obtain the y value of the vertex.
MAKE A TABLE
using x values close to the line of symmetry.
USE the equation
2bxa
-=
STEP 1: Find the line of symmetry
Let's Graph ONE! Try …
y = 2x2 – 4x – 1
( )4
12 2 2
bx
a
-= = =
A Quadratic Function in Standard Formy
x
Thus the line of symmetry is x = 1
Let's Graph ONE! Try …
y = 2x2 – 4x – 1
STEP 2: Find the vertex
A Quadratic Function in Standard Formy
x
( ) ( )22 1 4 1 1 3y = - - =-
Thus the vertex is (1 ,–3).
Since the x value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y value of the vertex.
*Since a is positive, we know the parabola opens up
5
–1
Let's Graph ONE! Try …
y = 2x2 – 4x – 1
( ) ( )22 3 4 3 1 5y = - - =
STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.
A Quadratic Function in Standard Formy
x
( ) ( )22 2 4 2 1 1y = - - =-
3
2
yx
Warm-Up ExercisesEXAMPLE 4 Find the minimum or maximum value
Tell whether the function y = 3x2 – 18x + 20 has a minimum value or a maximum value. Then find the minimum or maximum value.SOLUTION
Because a > 0, the function has a minimum value. To find it, calculate the coordinates of the vertex.
x = – b 2a = –
(– 18) 2a = 3
y = 3(3)2 – 18(3) + 20 = –7
ANSWER
The minimum value is y = –7.
Warm-Up ExercisesGUIDED PRACTICE for Examples 4 and 5
7. Find the y value of the vertex and tell whether It’s a minimum or maximum
y = 4x2 + 16x – 3.
x = – b 2a
= – 16
2(4)= –
16 8
= -2
y = 4(-2)2 + 16(-2) – 3
y = 16 – 32 – 3
y = -19
-19 minimum
Classwork Assignment: WS 4.1
(1-21 odd, 22-26 all)
