1. 10.2
Quadratic Functions

2. 10.2 – Quadratic Functions Goals / “I can…”
I can graph quadratic functions of the form y = ax + bx + c
I can graph quadratic inequalities 2

3. 10.2 – Quadratic Functions 2
Yesterday we learned about y = ax and y = ax + c. The a changes the ????? and the c moves the parabola ?????. 2

4. 10.2 – Quadratic Functions 2
Today we look at y = ax + bx + c. The bx moves the parabola horizontally, changing the location of the line of symmetry.

5. 10.2 – Quadratic Functions 2
In an equation y = ax + bx + c, the line of symmetry can be found by the equation
The x-coordinate of the vertex is

6. Finding the Line of Symmetry
10.2 – Quadratic Functions
Finding the Line of Symmetry
When a quadratic function is in standard form
For example…
Find the line of symmetry of y = 3x2 – 18x + 7
y = ax2 + bx + c,
The equation of the line of symmetry is
Using the formula…
Thus, the line of symmetry is x = 3
This is best read as …
the opposite of b divided by the quantity of 2 times a.

7. 10.2 – Quadratic Functions Finding the Vertex y = –2x2 + 8x –3
We know the line of symmetry always goes through the vertex.
STEP 1: Find the line of symmetry
STEP 2: Plug the x – value into the original equation to find the y value.
Thus, the line of symmetry gives us the x – coordinate of the vertex.
y = –2(2)2 + 8(2) –3
y = –2(4)+ 8(2) –3
y = –8+ 16 –3
To find the y – coordinate of the vertex, we need to plug the x – value into the original equation.
y = 5
Therefore, the vertex is (2 , 5)

8. A Quadratic Function in Standard Form
10.2 – Quadratic Functions
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.
MAKE A TABLE
using x – values close to the line of symmetry.
Plug in the line of symmetry (x – value) to obtain the y – value of the vertex.
USE the equation
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.

9. 10.2 – Quadratic Functions
Graph the equation
y = -3x + 6x + 5 2

10. 10.2 – Quadratic Functions Graphing Quadratic Inequalities
Graphing a parabola with an inequality is similar to a line. We use solid and dashed lines and we shade above (greater) or below (less) the line.

11. Forms of Quadratic Inequalities y<ax2+bx+c y>ax2+bx+c y≤ax2+bx+c y≥ax2+bx+c
Graphs will look like a parabola with a solid or dotted line and a shaded section.
The graph could be shaded inside the parabola or outside.

12. 10.2 – Quadratic Functions 1. Sketch the parabola y=ax2+bx+c
(dotted line for < or >, solid line for ≤ or ≥)
** remember to use 5 points for the graph!
2. Choose a test point and see whether it is a solution of the inequality.
3. Shade the appropriate region.
(if the point is a solution, shade where the point is, if it’s not a solution, shade the other region)

13. 10.2 – Quadratic Functions Graph y ≤ x2 + 6x - 4
Test point
Graph y ≤ x2 + 6x - 4
* Opens up, solid line
* Vertex: (-3,-13)
Test Point: (0,0)
0≤02+6(0)-4
0≤-4
So, shade where the point is NOT!

14. 10.2 – Quadratic Functions Graph: y > -x2 + 4x - 3
Test Point
Graph: y > -x2 + 4x - 3
* Opens down, dotted line.
* Vertex: (2,1)
x y 1 -3
* Test point (0,0)
0>-02+4(0)-3
0>-3

15. 10.2 – Quadratic Functions 1) y ≥ x2 and 2) y ≤ -x2 + 2x + 4 SOLUTION!
Graph both on the same coordinate plane. The place where the shadings overlap is the solution.
Vertex of #1: (0,0)
Other points: (-2,4), (-1,1), (1,1), (2,4)
Vertex of #2: (1,5)
Other points: (-1,1), (0,4), (2,4), (3,1)
* Test point (1,0): doesn’t work in #1, works in #2.

16. 10.2 – Quadratic Functions
Graph:
y > x + x+ 1 2