1. Graphing Quadratic Functions
y = ax2 + bx + c

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3. Quadratic Functions The graph of a quadratic function is a parabola.y x
The graph of a quadratic function is a parabola.
Vertex
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.
Vertex
NOTE: if the parabola opened left or right it would not be a function!

4. Standard Form a > 0 a < 0y x
The standard form of a quadratic function is
a < 0
a > 0
y = ax2 + bx + c
The parabola will open up when the a value is positive.
The parabola will open down when the a value is negative.

5. The line of symmetry ALWAYS passes through the vertex.
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.
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.
The line of symmetry ALWAYS passes through the vertex.

6. 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…
This is best read as …
the opposite of b divided by the quantity of 2 times a.
Thus, the line of symmetry is x = 3.

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

8. 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. A Quadratic Function in Standard Form
Let's Graph ONE! Try …
y = 2x2 – 4x – 1 y x
STEP 1: Find the line of symmetry
Thus the line of symmetry is x = 1

10. A Quadratic Function in Standard Form
Let's Graph ONE! Try …
y = 2x2 – 4x – 1 y x
STEP 2: Find the vertex
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.
Thus the vertex is (1 ,–3).

11. A Quadratic Function in Standard Form
Let's Graph ONE! Try …
y = 2x2 – 4x – 1 y x
STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. 3 2 y x –1 5

12. Notice, a is positive, so the graph opens up.
Problem 1
Notice, a is positive, so the graph opens up. y x
The vertex is at (2,-4)

13. Notice, a is negative, so the graph opens down.
Problem 2
Notice, a is negative, so the graph opens down. x y
The vertex is at (0,3)

14. Notice, a is positive, so the graph opens up.
Problem 3
Notice, a is positive, so the graph opens up. x y
The vertex is at (3,-5)

15. Solving a Quadratic The number of real solutions is at most two.
The x-intercepts (when y = 0) of a quadratic function
are the solutions to the related quadratic equation.
The number of real solutions is at most two.
Remind students that x-intercepts are found by setting y = 0 therefore the related equation would be ax2+bx+c=0. Also state that since the highest degree of a quadratic is 2, then there are at most 2 solutions. For the first graph ask “why are there no solutions?”-- there are no solutions because the parabola does not intercept the x-axis. 2nd and 3rd graph ask students to state the solutions. Additional Vocab may be itroduced: The x-intercepts are solutions, zero’s or roots of the equation.
One solution
X = 3
Two solutions
X= -2 or X = 2
No solutions