graphing quadratics
TRANSCRIPT
Graphing Quadratic Functions
MA.912.A.2.6 Identify and graph functions
MA.912.A.7.6 Identify the axis of symmetry, vertex, domain, range, and intercept(s) for a given parabola
Quadratic Function
y = ax2 + bx + c
Quadratic Term Linear Term Constant Term
What is the linear term of y = 4x2 – 3? 0x What is the linear term of y = x2- 5x ? -5x What is the constant term of y = x2 – 5x? 0 Can the quadratic term be zero? No!
Quadratic FunctionsThe graph of a quadratic function is a:
A parabola can open up or down.
If the parabola opens up, the lowest point is called the vertex (minimum).
If the parabola opens down, the vertex is the highest point (maximum).
NOTE: if the parabola opens left or right it is not a function!
y
x
Vertex
Vertex
parabola
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
a 0
y
x
Axis of Symmetry
Axis of Symmetry
Parabolas are symmetric.
If we drew a line down the middle of the parabola, we could fold the parabola in half.
We call this line the Axis of symmetry.
The Axis of symmetry ALWAYS passes through the vertex.
If we graph one side of the parabola, we could REFLECT it over the Axis of symmetry to graph the other side.
Find the Axis of symmetry for y = 3x2 – 18x + 7
Finding the Axis of SymmetryWhen a quadratic function is in standard form
the equation of the Axis of symmetry is
y = ax2 + bx + c,
2ba
x
This is best read as …
‘the opposite of b divided by the quantity of 2 times a.’
18
2 3x 18
6 3
The Axis of symmetry is x = 3.
a = 3 b = -18
Finding the VertexThe Axis of symmetry always goes through the _______. Thus, the Axis of symmetry gives us the ____________ of the vertex.
STEP 1: Find the Axis of symmetry
Vertex
Find the vertex of y = -2x2 + 8x - 3
2ba
x a = -2 b = 8
x 8
2( 2)
8
4 2
X-coordinate
The x-coordinate of the vertex is 2
Finding the Vertex
STEP 1: Find the Axis of symmetry
STEP 2: Substitute the x – value into the original equation to find the y –coordinate of the vertex.
8 8 22 2( 2) 4
ba
x
The vertex is (2 , 5)
Find the vertex of y = -2x2 + 8x - 3
y 2 2 2 8 2 3
2 4 16 3
8 16 3
5
Graphing a Quadratic Function
There are 3 steps to graphing a parabola in standard form.
STEP 1: Find the Axis of symmetry using:
STEP 2: Find the vertex
STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve.
MAKE A TABLE
using x – values close to the Axis of symmetry.
2ba
x
STEP 1: Find the Axis of symmetry
( )4
12 2 2
bx
a
-= = =
y
x
Graph : y 2x 2 4x 1
Graphing a Quadratic Function
STEP 2: Find the vertex
Substitute in x = 1 to find the y – value of the vertex.
( ) ( )22 1 4 1 1 3y = - - =-
Vertex : 1, 3
x 1
5
–1
( ) ( )22 3 4 3 1 5y = - - =
STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve.
y
x
( ) ( )22 2 4 2 1 1y = - - =-
3
2
yx
Graphing a Quadratic Function
Graph : y 2x 2 4x 1
y
x
Y-intercept of a Quadratic Function
y 2x 2 4 x 1 Y-axis
The y-intercept of aQuadratic function canBe found when x = 0.
y 2x 2 4x 1
2 0 2 4(0) 1
0 0 1
1
The constant term is always the y- intercept
The number of real solutions is at most two.
Solving a Quadratic
No solutions
6
4
2
-2
5
f x = x2-2 x +56
4
2
-2
5
2
-2
-4
-5 5
One solution
X = 3
Two solutions
X= -2 or X = 2
The x-intercepts (when y = 0) of a quadratic function are the solutions to the related quadratic equation.
Identifying Solutions
4
2
-2
-4
5
X = 0 or X = 2
Find the solutions of 2x - x2 = 0
The solutions of this quadratic equation can be found by looking at the graph of f(x) = 2x – x2
The x-intercepts(or Zero’s) of f(x)= 2x – x2
are the solutions to 2x - x2 = 0