1. QUADRATICS

2. The standard form of a quadratic equation is
y = ax² + bx + c

3. What is a quadratic? A more familiar word for quadratic is parabola
This is a function that we can graph
We can tell if a function is a quadratic if it is a polynomial equation of degree 2

4. Quadratic or not? y = 1 - x² y = 2(x – 3)(x + 4) x³ - y = 4

5. y = Ax² + Bx + C
Write the A, B, and C values for the functions that were quadratics
y = 1 - x²
y = 2(x – 3)(x + 4)
y + x² = 3x – 1
(x – 3)² = y + 2

6. The most basic form of a quadratic is y = x²
In this case, A = 1, B = 0, and C = 0
For all quadratics, A ≠ 0
Why?
What would this look like if we graphed it?

7. Table of Values
One method we have for graphing parabolas is by using a table of values
This allows us to plot points on a graph and estimate or sketch what we think the graph will look like

8. Plotting Points for y = x²
The most basic way for graphing a quadratic is to make a table of values x y -2 -1 1 2

9. Plotting Points for y = x²

10. Plotting Points for y = x² + 1
A = B = C = x y

11. y = x² + 1

12. Plotting Points for y = x² -2x + 3
A = B = C = x y

13. y = x² -2x + 3

14. Plotting Points for y = -2x² + x – 1
A = B = C = x y

15. y = -2x² + x – 1

16. Plotting Points for y = 5x² + 2x – 6
A = B = C = x y

17. y = 5x² + 2x – 6

18. Plotting Points for y = 0.5x² - 4x
A = B = C = x y

19. y = 0.5x² - 4x

20. Comparing to y = x²
In the previous examples, our ‘a’ value can tell us if our parabola opens up or down
If ‘a’ is positive, our parabola opens UP or is called concave up
If ‘a’ is negative, our parabola opens DOWN or is called concave down

21. If our parabola is concave up
This means our A value is positive
We would say our parabola has a minimum value at its lowest point

22. If our parabola is concave down
This means our A value is negative
We would say our parabola has a maximum value at its highest point

23. Example
y + x² = 3
‘a’ =
‘b’ =
‘c’ =
Parabola opens _______ x y

24. Example
3x² - 2x = 3 + y
‘a’ =
‘b’ =
‘c’ =
Parabola opens _______ x y

25. Example
x² = y – 1 + x
‘a’ =
‘b’ =
‘c’ =
Parabola opens _______ x y

26. Example y = -0.5x² + 2x – 1 ‘a’ = ‘b’ = ‘c’ = Parabola opens _______ x

27. Example
y + -2x² = x
‘a’ =
‘b’ =
‘c’ =
Parabola opens _______ x y

28. How else can we describe our parabola?
So far from the graph, we have talked about our parabola opening up or down
When describing parabolas from a graph, we can also talk about the axis of symmetry
This tells us the equation of the vertical line that divides our parabola evenly in half
Or more mathematically, it is the vertical line which divides the parabola so that one half is perfectly reflected or mirrored onto the other half
Let’s look at our previous examples and find axis of symmetry

29. Example y = 2x² - x + 3 ‘a’ = ‘b’ = ‘c’ = Parabola opens _______
Axis of Symmetry is ______

30. Example y = -3x² + 2x – 1 ‘a’ = ‘b’ = ‘c’ = Parabola opens _______
Axis of Symmetry is ______

31. Example y = -2x² - 3x + 1 ‘a’ = ‘b’ = ‘c’ = Parabola opens _______
Axis of Symmetry is ______