1. Graphing Quadratic Functions (10.1)
Objective:
Students will sketch the graph of a quadratic
function.

2. Algebra Standards:
16.0 Students understand the concepts of a relation and a
function, determine whether a given relation defines a
function, and give pertinent information about given relations
and functions.
21.0 Students graph quadratic functions and know that their
roots are the x-intercepts.
23.0 Students apply quadratic equations to physical problems,
such as the motion of an object under the force of gravity.

3. Vocabulary
Quadratic Function:
is a function that can be written in the
Standard form y = ax2 + bx + c, where a ≠ 0
Parabola:
is the U-shaped graph of a quadratic
Function.

4. is the lowest point on the graph of a parabola
Vocabulary
Vertex:
is the lowest point on the graph of a parabola
opening up or the highest point on the graph of
parabola opening down.
is the vertical line passing through
the vertex of a parabola and divides
the parabola into two symmetrical
parts that mirror images of each other.
Axis of
Symmetry

5. -a -c Vocabulary Quadratic Equation Quadratic Term Linear Term
Constant
Term a c +a
opens up
y-intercept -a
opens down +c
shifts up -c
skinny parabola
shifts down
wide parabola

6. Graphing a Quadratic Function
To graph y = ax2 + bx + c, where a ≠ 0, is a parabola.
Step 1: Determine the values of a, b, and c from the equation
Step 2: Determine if it opens up (+a) or down up(-a). b –
Step 3: Find the Axis of Symmetry, x = 2a
Step 4: Make a t-table, using x-values to the left and
right of the Axis of Symmetry
Step 5. Plot the Points

7. b – – – 2a 2(1) 2 Step 1: a = 1, b = 0, c = 0 Step 2:
#1 Graph a quadratic Function
Sketch the graph of y = x2
Step 1:
a = 1,
b = 0,
c = 0
Step 2:
Since a = +1, therefore it opens up. b – – –
Step 3:
A . S = = = = 2a
2(1) 2
Step 4: x
y = x2 y
(x, y) -3
(-3)2 9
(-3, 9) -2
(-2)2 4
(-2, 4)
Vertex = (0, 0) -1
(-1)2 1
(-1, 1) 02
(0, 0)
Axis of Symmetry
x = 0 1 12 1
(1, 1) 2 22 4
(2, 4) 3 32 9
(3, 9)

8. Step 5: Vertex = (0, 0) Opens up Axis of Symmetry x = 0 (x, y)
#1 Graph a quadratic Function
Sketch the graph of y = x2
Step 5:
Opens up x y
Vertex = (0, 0)
Axis of Symmetry
x = 0
(x, y)
(-3, 9)
(-2, 4)
(-1, 1)
(0, 0)
(1, 1)
(2, 4)
(3, 9)

9. b – – – 2a 2(1) 2 Step 1: a = 1, b = 0, c = 2 Step 2:
#2 Graph a quadratic Function
Sketch the graph of y = x2 + 2
Step 1:
a = 1,
b = 0,
c = 2
Step 2:
Since a = +1, therefore it opens up. b – – –
Step 3:
A. S (x)= = = = 2a
2(1) 2
Step 4: x
y = x2+ 2 y
(x, y) -3
(-3)2 + 2 11
(-3, 11) -2
(-2)2 + 2 6
(-2, 6)
Vertex = (0, 2) -1
(-1)2 + 2 3
(-1, 3)
02 + 2 2
(0, 2)
Axis of Symmetry
x = 0 1
12 + 2 3
(1, 3) 2 6
(2, 6) 3 11
(3, 11)

10. Step 5: Vertex = (0, 2) Opens up Axis of Symmetry x = 0 (x, y)
#2 Graph a quadratic Function
Sketch the graph of y = x2 + 2
Step 5:
Opens up x y
Vertex = (0, 2)
Axis of Symmetry
x = 0
(x, y)
(-3, 11)
(-2, 6)
(-1, 3)
(0, 2)
(1, 3)
(2, 6)
(3, 11)

11. Vertex = (0, -4) Opens up Axis of Symmetry x = 0
#3 Graph a quadratic Function
Sketch the graph of y = x2 – 4 x y
Opens up
Vertex = (0, -4)
Axis of Symmetry
x = 0

12. b 4 4 – – – 1 2a 2(-2) -4 Step 1: a = -2, b = 4, c = 1 Step 2:
#4 Graph a quadratic Function
Sketch the graph of y = -2x2 + 4x + 1
Step 1:
a = -2,
b = 4,
c = 1
Step 2:
Since a = -2, therefore it opens down. b 4 4 – – –
Step 3:
A.S (x)= = = = 1 2a
2(-2) -4

13. Step 4: x y = -2x2 + 4x + 1 y (x, y) -2 -1 Vertex = (1, 3) 1 2 3 4
#4 Graph a quadratic Function
Step 4: x
y = -2x2 + 4x + 1 y
(x, y) -2
-2• (-2)2 + 4(-2) +1
-15
(-2, -15) -8
– 8
+ 1
-2• (-1)2 + 4(-1) +1 -5 -1
(-1, -5) -2
– 4
+ 1
-2• (0)2 + 4(0) + 1 1
(0, 1)
+ 0
+ 1
Vertex = (1, 3) 1
-2• (1)2 + 4(1) +1 3
(1, 3) -2
+ 4
+ 1
Axis of Symmetry
x = 1 2 1
(2, 1) 3 -5
(3, -5) 4
-15
(4, -15)

14. Step 5: Vertex = (1, 3) Opens down Axis of Symmetry x = 1 (x, y)
#4 Graph a quadratic Function
Sketch the graph of y = -2x2 + 4x + 1
Step 5:
Opens down x y
Vertex = (1, 3)
Axis of Symmetry
x = 1
(x, y)
(-2, -15)
(-1, -5)
(0, 1)
(1, 3)
(2, 1)
(3, -5)
(4, -15)

15. Finding Critical Features of Quadratics
Vertex
Axis of Symmetry
Opens Up/Down
Opens Up
Y-intercept
(0, 2)

16. Find the critical features of the quadratic below.
Vertex
Axis of Symmetry
Opens Up/Down
Opens Down
Y-intercept
(0, 0)

17. Find the critical features of the quadratic below.
Vertex
Axis of Symmetry
Opens Up/Down
Opens Up
Y-intercept
(0, 4)