1. Finding the Zeroes

2. Review
Standard Form of a Quadratic Equation
y = Ax² + Bx + C

3. y = Ax² + Bx + C
We used standard form to create a table of values to be able to graph a quadratic
The ‘a’ value from standard form told us if our parabola was concave up or concave down
From our graph, we could also find the line that split our parabola evenly in half
We called this line the Axis of Symmetry

4. Sometimes it is not always easy or convenient to create a table of values to graph a quadratic
Instead, we may want to find where our parabola is crossing the x – axis and the y – axis
We call these points the intercepts
This can be of much more help when graphing

5. Today we want to look at
Where does the parabola go through the y-axis? (Y INTERCEPT)
Where does the parabola go through the x-axis? (X INTERCEPT)
How many times does the parabola cross the x-axis?

6. Finding y- intercepts
To find the y-intercept of any function, we set our x-value = 0 and solve our equation for y

7. The y-axis Find the y-intercept of y = -x² - 2x + 3 Step 1: Set x = 0
Step 2: Solve for y

8. Find the y-intercept y = 3x² + x – 2 Step 1: Set x = 0
Step 2: Solve for y
y = 3x² + x – 2

9. Find the y-intercept 9 + y = 0.8x² + 3x Step 1: Set x = 0
Step 2: Solve for y
9 + y = 0.8x² + 3x

10. Find the y-intercept y – 4 + x = -8x² Step 1: Set x = 0
Step 2: Solve for y
y – 4 + x = -8x²

11. Can we notice a pattern for finding the y – intercept when the quadratic is given to us in standard form?
For quadratics in the form y = ax² + bx + c, the y-intercept is always (0, c)
Let’s look at one graphically

12. Axis of Symmetry:
Direction of Opening
a =
Y-intercept =

13. The x-axis
Another thing we want to know about parabolas is where they cross the x-axis
The points where a parabola crosses the x-axis are called the x-intercepts.
When talking about quadratics, we also call these points the zeroes, roots, or solutions

14. Finding x - intercepts
To find x-intercepts, we use the same method as finding y-intercepts except we set y = 0
y =
What do we do from here?
Are there methods we know for solving this equation?

15. Finding x – intercepts
To find x – intercepts of a quadratic equation, we want to convert it into the form we know as intercept form

16. Intercept Form
We can convert quadratics in standard form into what we call intercept-form
This form will show us where the graph crosses the x-axis
We can rewrite our equation as y = a(x – s)(x – t)

17. How can we change Ax² + Bx + C to A(x – s)(x – t) ?
We know this method better as factoring

18. y = a(x – s)(x – t)
If we FACTOR our quadratic equation into two brackets, it will give us the values where our parabola crosses the x-axis
In this intercept form, we can say our quadratic crosses the axis at x = s and x = t
We call this method Finding the Zeroes, Finding the Roots, or Finding the Solutions

19. Steps for Solving Quadratics
Step 1. Set y = 0
Step 2. Factor the equation into intercept form
Step 3. Set each set of brackets = 0
Step 4. Solve each equation

20. Examples in Intercept Form
Find the x – intercepts of the following quadratics
y = (x – 2)(x + 3)
y = (x + 4)(x – 1)
y = (2x + 3)(3x – 4)
Step 1. Set y = 0 Step 2. Factor the equation into intercept form Step 3. Set each set of brackets = 0 Step 4. Solve each equation

21. What are the x-intercepts of the following quadratics in standard form?
Set y = 0, solve for x
y = x² + 5x + 6
y = x² - 4x + 4
Step 1. Set y = 0 Step 2. Factor the equation into intercept form Step 3. Set each set of brackets = 0 Step 4. Solve each equation

22. What are the x-intercepts of the following quadratics in standard form?
Set y = 0, solve for x
y = 7x² + x – 8
y = 14 – x – 3x²
Step 1. Set y = 0 Step 2. Factor the equation into intercept form Step 3. Set each set of brackets = 0 Step 4. Solve each equation

23. There are three possibilities
A quadratic function has three possibilities for number of zeroes
1. No Zeroes
2. One Zero
3. Two Zeroes

24. No Zeroes
In what cases will a quadratic function have no zeroes?

25. One Zero
In what cases will a quadratic function have one zero?

26. Two Zeroes
In what case will a quadratic function have two zeroes?

27. Axis of Symmetry:
Zeroes:
Direction of Opening
Y-intercept =

28. Using our Intercepts to Graph
If we can plot our intercepts on a graph, we can use the axis of symmetry to help us complete the rest of the graph
The axis of symmetry will divide our two zeroes equally in half
This will be where our maximum or minimum point passes through
Let’s look at some examples

29. y = -2x² + 8x + 8
y = x² + 5x + 6

30. y = -0.5x² + x – 1.5
y = x² - 2x - 24

31. y = -2x² + 8x + 8 y = x² + 5x + 6 y = -0.5x² + x – 1.5
We can find the axis of symmetry by adding our two zeroes and dividing it by two.
This is finding the center which is where our AOS passes through
y = -2x² + 8x + 8
y = x² + 5x + 6
y = -0.5x² + x – 1.5
y = x² - 2x - 24